Wikiversity:Colloquium

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== Technical Request:  Courtesy link.. ==

[[Template_talk:Information#Background_must_have_color_defined_as_well]] [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
: I can't edit the template directly as it need an sysop/interface admin to do it. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
:: Also if the Template field of - https://en.wikiversity.org/wiki/Special:LintErrors/night-mode-unaware-background-color is examined, there is poential for an admin to clear a substantial proportion of these by implmenting a simmilar fix to the indciated templates (and underlying stylesheets). It would be nice to clear things like Project box and others, as many other templates (and thus pages depend on them.) :)
[[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)

:I think it would be best to grant you interface admin rights for a short period of time to make these changes. However, I still have doubts about the suitability of this solution, which may cause other problems and no one has explained to me why dark mode has to be implemented this way @[[User:ShakespeareFan00|ShakespeareFan00]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:43, 20 March 2026 (UTC)

: I would have reservations about holding such rights, which is why I was trying to do what I could without needing them. However if it is the only way to get the required changes made, I would suggest asking on Wikipedia to find technical editors, willing to undertake the changes needed. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:32, 21 March 2026 (UTC)

== WikiEducator has closed ==

Some of you may know of a similar project to Wikiversity, called [https://wikieducator.org/Main_Page WikiEducator], championed by [https://oerfoundation.org/about/staff/wayne-mackintosh/ Wayne Mackintosh][https://www.linkedin.com/posts/waynemackintosh_important-notice-about-the-oer-foundation-activity-7405113051688931329-Nhm9/][https://openeducation.nz/killed-not-starved/].

It seems [https://openeducation.nz/terminating-oer-foundation their foundation has closed] and they are no longer operating.

They had done quite a bit of outreach (e.g., in the Pacific and Africa) to get educators using wiki.

The WikiEducator content is still available in MediaWiki - and potentially could be imported to Wikiversity ([https://wikieducator.org/WikiEducator:Copyrights CC-BY-SA] is the default license).

The closing of WikiEducator arguably makes the nurturing of Wikiversity even more important.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:09, 1 April 2026 (UTC)

:I was never active there. If anyone has an account or is otherwise in contact, we may want to copy relevant information here or even at [[:outreach:]]. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:46, 1 April 2026 (UTC)

:: I reached out to [[User:Mackiwg~enwikiversity|Wayne]] in January, and he responded briefly but positively (while travelling).  I wrote to the low-traffic wikieducator mailing list today and got a nice [https://groups.google.com/g/wikieducator/c/r_yIyUw6ZIA reply] from [[user:SteveFoerster|Steve Foerster]] who's interested in helping. If we can figure out a migration path it would be great to adopt at least the main namespace pages here.  
:: A few questions that come to mind: 
:: - would people want to create matching user accounts
:: - are there any namespaces (user/talk?) that should not be moved over
:: We could look at how this was done for the [[m:Wikivoyage/Migration]] wikivoyage migration. &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 04:27, 1 May 2026 (UTC)
:::That's fantastic, SJ, that you've reached out and that Wayne, Steve, and Jim are receptive—and that you can help! -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:52, 1 May 2026 (UTC)
::::A matching accound makes sense to give credits to the original authors and keep a clean chain of versions. The initial commit into wikiversity could have a &quot;marker with timestamp&quot; similar to signature with a reference where the content's source or a Web archive. This would allow authors to continue there work on wikiversity if they wish. [[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 06:30, 15 May 2026 (UTC)

== Wikinews is ending ==

Apparently mainly due to low editorial activity, low public interest, but also failure to achieve the goals from the proposal for the creation of the project, the Wikinews project is ending after years of discussions ([[Meta:Proposal for Closing Wikinews|some reading]]).

And I would be interested to see how Wikiversity is doing in the monitored metrics. We probably have more editors than Wikinews had, but what about consumers and achieving the goals? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:14, 1 April 2026 (UTC)

:Wikiversity's biggest issue in recent times was the hosting of low-quality, trash content. Thankfully we've done a great job in removing pseudoscience and other embarrassingly trash content (Wikidebates, for example), but the biggest concern moving forward is proper maintenance IMO. I've caught several pseudoscience pages being created within the last few months that could easily have flown under the radar (ex, [[The Kelemen Dilemma: Causal Collapse and Axiomatic Instability]]), so I'd urge our custodians/curators to be on the lookout for this type of content. Usually an AI-overview can point this type of content out relatively well.
:In terms of visibility, I believe Wikiversity is a high-traffic project. I remember my [[Mathematical Properties]] showing up on the first page of Google when searching up &quot;math properties&quot; for the longest time (and is still showing up in the first page 'till this day!). Besides, Wikinews hosted a lot of short-term content (the nature of news articles), while Wikiversity hosts content that can still be useful a decade later (ex, [[A Reader's Guide to Annotation]]). 
:I think we are on a better path than we were a few months ago, and I do want to thank everyone here who has been helping out with maintaining our website! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:48, 1 April 2026 (UTC)
:For what it's worth, the group that did that study has since disbanded, so no one is monitoring the other sister projects in the same way. Additionally, Wikinews had some catastrophic server issues due to the maintenance of [[:m:Extension:DynamicPageList]] which don't apply here. Your questions are still worth addressing, but I just wanted to cut off any concern at the pass about Wikiversity being in the same precarious situation. Wikiversity is definitely the biggest &quot;lagging behind&quot; or &quot;failure&quot; project now that Wikinews is being shuttered, but I don't see any near- or medium-term pathway to closing Wikiversity. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:46, 2 April 2026 (UTC)

:[[w:en:Wikipedia:Wikipedia Signpost/2026-03-31/News and notes|Entirety of Wikinews to be shut down]] (Wikipedia Signpost) -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:03, 11 April 2026 (UTC)

== Add some user rights to the curator user group? ==

By default, only custodians have the ability to mark new pages as patrolled (&lt;code&gt;patrol&lt;/code&gt;) and have their own page creations automatically marked as patrolled (&lt;code&gt;autopatrol&lt;/code&gt;). I am proposing both of the following:
* Curators can mark new pages as patrolled, helping on reducing the backlog of new, unpatrolled pages.
* New pages made by curators will be automatically marked as patrolled by the MediaWiki software.
Before we implement this, I would suggest implementing a proposed guideline for marking new pages as patrolled for curators and custodians.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:32, 17 April 2026 (UTC)

:Agree, &lt;s&gt;also can we also allow curators to undelete pages since they already have the rights to delete them?&lt;/s&gt; [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:54, 18 April 2026 (UTC)
::I think the requirement that undelete NOT be included came from above (meta / stewards / central office). Having access to the undelete page gives access to information that is restricted by their policies to admins (custodians and bureaucrats). -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 20:12, 18 April 2026 (UTC)
::: [[User:PieWriter|PieWriter]], unless if requests for curator and custodian should be RfA-like processes (that is, including voting and comments), then I have to agree with Dave above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:03, 18 April 2026 (UTC)
::::Oh, I didn’t realise that. Withdrawing my comment.. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:08, 19 April 2026 (UTC)
:{{support}} Seems reasonable and would reduce overhead. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 14:35, 18 April 2026 (UTC)
:'''Agree''', implement it also to [[Wikiversity:Curators]] proposal please. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:11, 18 April 2026 (UTC)
: I went ahead and filed [[phab:T424445]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:39, 26 April 2026 (UTC)
: This was completed on 30 April.
: Perhaps we could benefit from some documentation (e.g., [[Wikiversity:Patrol]] or [[Wikiversity:Patrolling]]?) and updates to the curator, custodian, bureaucrat pages? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:49, 19 May 2026 (UTC)
:: Yes, but I would recommend [[Wikiversity:Patrolling]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 02:26, 19 May 2026 (UTC)

== [[Wikiversity:Curators|Curators and curators policy]] ==
{{archive top|There is strong consensus, so [[Wikiversity:Curators]] is now a policy. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:15, 9 May 2026 (UTC)}}
How does it come, that Wikiversity has curators, but Curators policy is still being proposed? How do the curators exists and act if the policy about them havent been approved yet? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:33, 16 October 2025 (UTC)

:It looks as if it is not just curators. The policy on Bureaucratship is still being proposed as well. See [[Wikiversity:Bureaucratship]]. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 18:33, 27 October 2025 (UTC)

:I think its just the nature of a small WMF sister project in that there are lots of drafts, gaps, and potential improvements. In this case, these community would need to vote on those proposed Wikiversity staff policies if we think they're ready. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:08, 3 December 2025 (UTC)
:What? I thought you were getting it approved, Juandev... :) [[User:I&amp;#39;m Mr. Chris|I&amp;#39;m Mr. Chris]] ([[User talk:I&amp;#39;m Mr. Chris|discuss]] • [[Special:Contributions/I&amp;#39;m Mr. Chris|contribs]]) 14:20, 12 February 2026 (UTC)
::Yeah I think this one is important too and we need to aprove it too @[[User:I'm Mr. Chris|I'm Mr. Chris]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 12 February 2026 (UTC)
:::I thinks its ready to made into a policy, it seems to be complete and informative about what the rights does and how to get it. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:08, 15 February 2026 (UTC)
::::Agree -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:00, 27 March 2026 (UTC)

Let's make this the official discussion about adopting the [[Wikiversity:Curators|curators policy]] policy. Your comments are invited and welcome. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:40, 24 April 2026 (UTC)
: There were two similar Colloquium threads in separate places about the proposed curators policy. So I've moved them to be adjacent. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:42, 1 May 2026 (UTC)
{{archive bottom}}

== Wikiversity:Curators to become a policy ==
{{archive top|There is strong consensus, so [[Wikiversity:Curators]] is now a policy. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:16, 9 May 2026 (UTC)}}
I've looked at the discussions about the Curators policy, I've looked at the practices, and it seems to me that there is no dispute about the wording of the policy, and what's more, the community has been using this proposal as if it were an offical policy for several years. Therefore, I propose that [[Wikiversity:Curators]] become a policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:35, 18 April 2026 (UTC)

:{{support}} —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 18 April 2026 (UTC)
:{{support}} —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:21, 18 April 2026 (UTC)
: {{support|Yes, please}}. Especially after when I and PieWriter proposed above, I agree. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:27, 24 April 2026 (UTC)
:: @[[User:Juandev|Juandev]]; as of now, curators now have the user rights &lt;code&gt;autopatrol&lt;/code&gt; and &lt;code&gt;patrol&lt;/code&gt;. Perhaps we should also include that in [[Wikiversity:Custodianship]]? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 12:07, 30 April 2026 (UTC)
:::You meant [[Wikiversity:Curators]] @[[User:Codename Noreste|Codename Noreste]]? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:15, 5 May 2026 (UTC)
:::: I agree that we must develop what rules curators should follow when marking new pages as patrolled; the same can be added for custodians since they can also mark new pages as patrolled. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:37, 5 May 2026 (UTC)
:::::I see, well I think you can just add this to the policy. It is not major change and it probably reflects actual practice or actual technical possibilities for those flags. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:20, 7 May 2026 (UTC)
:{{support}} -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:42, 1 May 2026 (UTC)
:{{Support}} per nom. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 13:32, 1 May 2026 (UTC)
{{Archive bottom}}

== Is anyone interested in Neurodiversity? ==

Is anyone interested in Neurodiversity? Is there anyone here who is interested for Neurodiversity to be &quot;something more&quot; than it already is? Does anyone here consider Neurodiversity one of the &quot;harder topics&quot; to work on or discuss? Does anyone here have an opinion about the [[Neurodiversity Movement]]? So these questions don't appear like &quot;out of a vacuum&quot; I can tell you a bit about my background:

Many years ago I got a psychiatric diagnosis &quot;Asperger's&quot;. After I stepped out of the office and my Äsperger's was 'concluded', I stepped out into the street and thought my first negative thought(but the positive thought followed after). The thought was about concentration camps in the second world war and that the world seemed to be going into the direction of &quot;labeling others&quot;. I was unsure whether this was &quot;real science&quot; and sort of &quot;challenged myself&quot; to make up my own mind after meeting people that had been given this diagnosis. The more adults with this diagnosis I met the more I started seeing &quot;patterns&quot;.

Was it a coincidence that the first person with Asperger's I met reminded me about my father later after I had plenty of times of experience with interacting with him? None of the people I interacted with online through IRC text chat...I felt I got any clue about how &quot;their brains work&quot;. Only when I met one person from the Asperger's chat community in person we both realized that whatever we experienced was akin to the &quot;chaos theory&quot;. He told me about &quot;chaos theory&quot; while I didn't know even what that term meant but I guess I 'read between the lines'. My question that I linger on still today is &quot;did he understand about me what I think I understood about him?&quot;? That our brains had the same configuration? Most autistic adults who meet other autistic adults usually get disappointed. They think the diagnosis will help them meet somebody like themselves and then they realize the great diversity in the autistic spectrum created by Psychiatry.

I later stopped interacting with autistic communities that much, I felt that it did not benefit me. Also Neurodiversity's &quot;neurotypes&quot; interested me for a while until I realized I had &quot;misunderstood everything&quot; about them and how they are used in the Neurodiversity Movement or &quot;Neurodiversity community&quot; if that even can precisely be defined? I doubt it but if you want to contribute to the [[Neurodiversity Movement]]. My previous attempts failed as I got more and more confused. I think a community project needs a community. With a lack of that I don't think it is worth my time. If any of you would like to work on that project let me know on my talk page.

So I was kinda lost and was talking to my friend and psychologist and I realized if I never talk about my idea to anyone in a &quot;comprehensive way&quot; or show that it matters to me nothing is going to ever happen. So I started talking about my &quot;idea&quot; more. Nobody could understand the &quot;idea&quot; because I had not developed my skills regarding where to start...although the process had already started &quot;automatically&quot; and that's why I often think of &quot;well my brain sort of activated me&quot;. I don't feel like I did have a plan and this idea happened. It happened &quot;by itself&quot;. My brain reacted to what I was seeing in a video or stream.

I value interaction highly in this idea. I think it would be helpful to make a community of people who are not paranoid about stuff that can express itself like &quot;don't analyze me!&quot;, &quot;don't compare me to anyone!&quot;. 

On the contrary, more often than not those adults who were diagnosed were actually openly comparing themselves with each other and I think that is healthy in a &quot;science&quot; way if done the &quot;right way&quot; which probably means &quot;Do no harm&quot;.

I found video material is important but I'm very unsure if uploading own video material to Wikimedia Commons would constitute a &quot;reasonable&quot; use of the resources there. Maybe somebody here needs to ask more questions to me that I should answer before that happens. I also know the '''be bold''' so I could just do what I think might be ok. Though I work better in a group as long as I know what &quot;group configurations&quot; help me. This is in a non-profit way. Since the state supported me this might be a way I am trying to &quot;give back&quot; to the state and &quot;the world&quot;. May seem overly ambitious and crazy but this thing gives me energy. It gives me hope when trying to develop this idea. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:47, 23 April 2026 (UTC)

:Thanks for sharing. There is plenty of room for neurodiversity community learning. However, the challenge I think is that the intersection of those interested in (a) ND, and (b) English Wikiversity might be very small (e.g., 1!) at this point in time.
:But don't give up hope. For example, Wikipedia has many more ND-interested editors; maybe consider reaching out to see who might be interested:
:[[w:Category:Wikipedians interested in neurodiversity]]
:You could also start an equivalent category here:
:[[:Category:Wikiversitarians interested in neurodiversity]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:46, 6 May 2026 (UTC)

== Request for comment (global AI policy) ==

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;A [[:m:Requests for comment/Artificial intelligence policy|request for comment]] is currently being held to decide on a global AI policy. {{int:Feedback-thanks-title}} [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 00:58, 26 April 2026 (UTC)&lt;/bdi&gt;
&lt;!-- Message sent by User:Codename Noreste@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=30424282 --&gt;

== Language learning  ==

toki! I am trying to add or see what the toki pona language learning stuff on here is but I don't see anything that is language learning for anything. [[User:Jan Imon|Jan Imon]] ([[User talk:Jan Imon|discuss]] • [[Special:Contributions/Jan Imon|contribs]]) 23:13, 2 May 2026 (UTC) —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:29, 3 May 2026 (UTC)

:We have language materials ([[:Category:Languages]], [[World Languages]], [[Portal:Foreign Language Learning]], [[Portal:Multilingual Studies]]). They are not as developed as I think we would all like and there's not any coverage of Toki Pona, but in principle, we could and would like that. You can also see [[:b:Subject:Languages]] at our sister project Wikibooks. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:33, 3 May 2026 (UTC)

== Timeline format? ==

I’ve been working on the World War II articles, including the [[World War II/Timeline|timeline]], and is there a specific timeline format that should be used? Right now it’s just a table, and there’s no separation between different periods/phases of the war.

I don’t want to use [[mw:Extension:EasyTimeline]] because this will be displaying dates and not time periods. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 01:35, 4 May 2026 (UTC)

:I dont think we have a policy or guideline, how to format a timeline. But you may try to browes wikiversity by Google if someone was dealing with this in the past somewhow @[[User:PhilDaBirdMan|PhilDaBirdMan]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:23, 5 May 2026 (UTC)
::+1 - there's no specific guideline on how to format a timeline, it's really up to you. In my opinion I think the timeline is good. I'd personally bold the dates just to make it easier to separate it from the event description, but that's my personal 2 cents. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:18, 5 May 2026 (UTC)
:::I’ll probably remove links to the dates/years, they’re just Wikipedia pages that shouldn’t be over linked to. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 00:39, 6 May 2026 (UTC)

== Interface administrator for Codename Noreste ==
{{Archive top|After running for a week, there is clear consensus for [[User:Codename Noreste]] to have Interface admin rights for 120 days; implemented until 10 September, 2026 -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:36, 12 May 2026 (UTC)}}

Hello, everyone. I am requesting interface administrator access on this wiki.

The main reasoning is that I would benefit from having the user right &lt;code&gt;editinterface&lt;/code&gt;, which would allow me to make dark mode changes to pages in the MediaWiki namespace, add &lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;mw-parser-output&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt; to some interface pages using templates, handle interface-protected edit requests, and similar stuff. Additionally, I have some knowledge of CSS, and I would like to assist with modifying CSS pages whenever necessary, such as moving MediaWiki common.css code to TemplateStyles CSS pages.

I am requesting the maximum time that is allowed per the [[Wikiversity:Interface administrators|policy]], and I have 2FA enabled on my account. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:55, 6 May 2026 (UTC)

*{{support}} Globally trusted user. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:07, 6 May 2026 (UTC)
*{{support}} Trusted and knowledgeable. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:35, 6 May 2026 (UTC)
*{{support}} WV would benefit from this. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 08:32, 6 May 2026 (UTC)
*{{support}} --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:13, 7 May 2026 (UTC)

:{{Comment}} Could @[[User:Codename Noreste|Codename Noreste]] delete [[MediaWiki:Gadget-WikiSign.js]], which was requested to be deleted @[[User:Koavf|Justin]], @[[User:Jtneill|Jtneill]], @[[User:Atcovi|Atcovi]]? I dont think we need it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:40, 9 May 2026 (UTC)
::Yes - clearly no longer used -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:18, 9 May 2026 (UTC)
::: I can't delete it because I don't have the required permissions to do so.
::: On a side note, if this project has a need for permanent interface administrators, I would suggest that we have a minimum of two IAs, similar to how there must be two CUs and/or suppressors (or none). Maybe Koavf can be a good candidate if I am elected for permanent interface adminship, and I believe that permission shouldn't be removed from someone's own account. Instead, a bureaucrat should do it. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:20, 9 May 2026 (UTC)
::::I am willing and happy to do it, unfortunately, we do not have an appetite for indef IAs and just had a discussion that resulted in a [https://en.wikiversity.org/w/index.php?title=Wikiversity:Interface_administrators&amp;diff=prev&amp;oldid=2807543 consensus that we can have IAs that have the user rights for 14 to 120 days]. So once you have the rights, please make sure to gopher it. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:54, 9 May 2026 (UTC)
:::::@[[User:Koavf|Koavf]] give it time. Look at me, I was in favor of shorter time, now I am looking back to times, when custodians could do it without the need of extra flag. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:31, 9 May 2026 (UTC)
::::::Here's hoping. I think it would reduce administrative overhead, but that's just me and I'm not a bureaucrat here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:33, 9 May 2026 (UTC)
::::Complicated. Where are the times, admins could do everything! [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:27, 9 May 2026 (UTC)
{{archive bottom}}

== [[Wikiversity:Bureaucratship]] to become a policy ==
{{archive top|'''Approved - now a policy'''. 5 supports + 1 nominator. No objections.}}
Following the recent approval of [[Wikiversity:Curators]] as a policy, I think [[Wikiversity:Bureaucratship]] may also be ready for policy status.

Please share your views about whether bureaucratship is ready to become a policy, or whether further revisions are needed.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 13:58, 9 May 2026 (UTC)
: I added a logo about that user group, but other than that, it looks good to me. {{support}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:38, 9 May 2026 (UTC)
:I think that the consensus on this policy is proven by years of using it without further changes. But I I have to say weather I agree with this to become a policy, than of course {{support}}. It works and there were no major issues with it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:45, 9 May 2026 (UTC)
:{{support}} no issues. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:51, 10 May 2026 (UTC)
:{{support}} [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:37, 11 May 2026 (UTC)
:{{support}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:51, 12 May 2026 (UTC)
{{archive bottom}}

== Reminder about custodian-related pages ==

I would like to remind the community about what the following custodian pages are:
* [[Wikiversity:Request custodian action]] is for requesting actions to be done by custodians, and
* [[Wikiversity:Notices for custodians]] is for notices of interest to custodians, like an administrator's noticeboard
Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:12, 12 May 2026 (UTC)

:Thanks - I needed this reminder :) -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:21, 12 May 2026 (UTC)

== [[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/&amp;#126;2026-28640-56|&amp;#126;2026-28640-56]] ([[User talk:&amp;#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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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)

== Create a pseudo-bot user group? ==

I would like to propose adding a new user group to Wikiversity: Pseudo-bot (&lt;code&gt;flood&lt;/code&gt;). This will allow users to perform repetitive actions without flushing the recent changes feed (with only the &lt;code&gt;bot&lt;/code&gt; user right). However, I would suggest that for the pseudo-bot user group:
* It can be granted and revoked by custodians. &lt;s&gt;However, can curators add and remove pseudo-bot from their own accounts (and not others)?&lt;/s&gt;
* Users can remove themselves from it.
* A guideline might be necessary about the information and usage of it.
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:31, 14 May 2026 (UTC)

:This sounds good. Which other wiki could we model this user group on? e.g., [[b:Wikibooks:Pseudo-bots]]? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:19, 15 May 2026 (UTC)
::@[[User:Jtneill|Jtneill]] Wikiquote has a similar group: [[:wikiquote:Special:ListGroupRights]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:25, 15 May 2026 (UTC)
: Should we allow curators to add and remove themselves from the pseudobot user group (from their own account) as well? I see no objections to creating the user group. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:20, 18 May 2026 (UTC)
::My thinking is perhaps not curators by default because there should be clear visibility about their actions until they are well trusted. Let's draft a guideline or proposed policy ([[Wikiversity:Pseudo-bots]]) for the proposed user group. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:39, 18 May 2026 (UTC)
::: A solution is that they can ask any custodian to grant that group, and to remove themselves when done. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:17, 19 May 2026 (UTC)
:::: Yes, that sounds good. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:12, 19 May 2026 (UTC)

== Coming over From wikinews ==
Any chance someone could help me if you are allowed to write news articles here since wikinews is going read only mode soon, thank you! [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:43, 1 May 2026 (UTC)

:The scope of Wikiversity is very broad and is basically about more-or-less any learning material. We have made it a point to not have duplicative content of other WMF projects, but since Wikinews is being shuttered, I personally am fine with writing news articles here. One thing that is not controversial at all is a learning resource &lt;em&gt;about&lt;/em&gt; how to write news: that could be hugely useful here and could involve the process of writing news stories to learn and to share back and forth with an editor or fact-checker. In fact, I'd support an entire namespace dedicated to keeping the notion of Wikinews alive here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:38, 1 May 2026 (UTC)
::Thank you so much! How do I start? Cheers! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:07, 2 May 2026 (UTC)
:::I think it's premature to start just making news articles en masse, but if you want to start discussing the topic of citizen journalism, you can do that now. [[:Category:Journalism]] already has some material, so you can start by seeing what we already have, how you can refine that, etc. You can definitely have learning resources with collaborators who want to learn about journalism ASAP. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:24, 2 May 2026 (UTC)
::::thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:38, 2 May 2026 (UTC)
::::If I could try and start one News Article could you please tell me how to go about it? Like what style of writing like Wikinews or something else? Thank you Justin! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:48, 2 May 2026 (UTC)
:::::Honestly, there are very few policies and guidelines here. I think the best way to write a news story would be in a manner that is obvious and instructive. So, for instance, it's common to use the &quot;pyramid style&quot; when you're writing news, so if you were to write a story that makes it very clear that you are using that approach, that would be helpful. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 02:08, 2 May 2026 (UTC)
::::::cool thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 02:13, 2 May 2026 (UTC)
::::::im ready to write @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 21:30, 13 May 2026 (UTC)
:::::::I think we should get more local consensus for a big project like including the entirety of the scope of Wikinews here. Again, I support it personally. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:55, 13 May 2026 (UTC)
::::::::ok lets begin. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:15, 13 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 &quot;practicing journalism&quot; 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)

===Comments and questions===
:Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice.
:Initial questions:
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
:* What are &quot;active editions&quot;?
:* 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 &quot;active editions&quot;?
::*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 &lt;em&gt;about&lt;/em&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)

== Inactivity policy for Curators ==

I was wondering if there is a specific inactivity polity for curators (semi-admins) as I am pretty sure the global policy does not apply to them as they are not ''fully'' sysops. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:20, 15 February 2026 (UTC)

:Unfortunately, I don't see an inactivity policy, but if we were to create such a new policy for curators, it should be the same for custodians (administrators). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:45, 15 February 2026 (UTC)
::@[[User:Codename Noreste|Codename Noreste]] There is currently none, that I could find, for custodians either. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:47, 17 February 2026 (UTC)
:::I think we should propose a local inactivity policy for custodians (and by extension, curators), which should be at least one year without any edits ''and'' logged actions. However, I don't know which page should it be when the inactivity removal procedure starts. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:53, 17 February 2026 (UTC)
::::@[[User:Codename Noreste|Codename Noreste]] In theory, there should be a section added at [[WV:Candidates for custodianship]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:55, 17 February 2026 (UTC)
::::: To be consistent with the [[meta:Admin activity review|global period of 2 years inactivity]] for en.wv  [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship#How are bureaucrats removed?|Bureaucrats]] we could add something like this to [[Wikiversity:Curators]]:
::::::The maximum time period of inactivity &lt;u&gt;without community review&lt;/u&gt; for curators is two years (consistent with  the [[:meta:Category:Global policies|global policy]] described at [[meta:Admin activity review|Admin activity review]] which applies for [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship|Bureaucrats]]). After that time a custodian will remove the rights.
::::: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:51, 27 March 2026 (UTC)
:::::Yup, I agree with Jtneill, there is a policy proposal for Wikiversity:Curators, where it should be logically deployed. The question is if we are ready to aprove the policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:43, 17 April 2026 (UTC)
:::::: I agree, but we should notify the colloquium about inactive curators, just like a steward would do for inactive custodians and bureaucrats per [[:m:Admin activity review|AAR]]. What is the minimum timeframe an inactive curator should receive so they can respond they would keep their rights? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:49, 17 April 2026 (UTC)
:I incorporated these suggestions into the proposed curators policy. Please review/comment/improve. Summary: 2 years, notify curator's user page, then remove rights after 1 month: [[Wikiversity:Curators#Inactivity]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:59, 24 April 2026 (UTC)
:: @[[User:Jtneill|Jtneill]] I created [[Template:Inactive curator]] for this. Feel free to make any changes or improvements. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:29, 24 April 2026 (UTC)
:::Wondering, should we also have:
:::* {{tl|Inactive custodian}}
:::* {{tl|Inactive bureaucrat}}
:::or perhaps just a single template with a parameter(s) for the user right(s)/role(s)? e.g., 
:::* if a custodian is inactive for 2 years, then custodian and curator rights are to be removed and 
:::* if a bureaucrat is inactive for 2 years, then bureaucrat, custodian, and curator rights are to to be removed.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:58, 13 May 2026 (UTC)
:::: I would probably modify that template when we actually develop our own inactivity policy, because we're currently under the AAR (a steward notifies the colloquium with [[m:Admin activity review/Notice to communities]], and inactive advanced right holders with [[m:Admin activity review/Notice to inactive right holders]]). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:16, 13 May 2026 (UTC)
:::::Ah, I see. Yes, that makes sense. Thankyou. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:21, 15 May 2026 (UTC)
: In that case, should we develop our own inactivity policy (e.g. on [[Wikiversity:Inactivity policy]] or [[Wikiversity:Support staff/Inactivity]])? I would list the general inactivity part, the process, etc. Once it's approved as a policy, I will [[m:Stewards' noticeboard|notify the stewards]]. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:30, 16 May 2026 (UTC)
::Originally, I would have thought that, for a small wiki like en.wv, it made sense to leave inactivity monitoring to the stewards. However, with the creation of the curator user group, we have already taken on local responsibility for monitoring inactivity in at least one advanced-rights group. Extending this to custodians and bureaucrats would not add much additional overhead and would provide a more consistent and transparent local administrative process.
::One option would be to develop a single, centralised policy covering all advanced-rights groups.
::An alternative would be to include an ==Inactivity== section on each relevant policy page (e.g., we already have [[Wikiversity:Curatorship#Inactivity]], but not yet in the custodianship, and bureaucratship policy pages). This approach would allow some flexibility because different user groups may warrant different criteria (such as inactivity thresholds, qualifying activity, or review procedures).
::A hybrid approach may be best: maintain separate inactivity sections within each user-group policy page, while transcluding these into a central overview page such as Codename Noreste suggests. This would preserve clarity at the local policy level while also providing a single reference point for consistency and oversight. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:09, 16 May 2026 (UTC)
::: I would suggest we develop a centralized inactivity policy page, and include a short summarized section of that page, on the support staff user group pages. We must also include a link to that policy page if we were to add &lt;nowiki&gt;== Inactivity ==&lt;/nowiki&gt; to each of those user group pages. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:48, 17 May 2026 (UTC)

== Inactive curators ==

Hello, even though [[Wikiversity:Curators]] is not a policy yet, there are curators listed here that have been inactive for two years or more:
* {{user|Cody naccarato}} (last edit on 13 Dec 2022, last logged action on 10 Dec 2022)
* {{user|Praxidicae}} (last edit on 10 Sep 2022, last logged action on 12 Sep 2022)
[[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 19 April 2026 (UTC)

:Yup, I would remove the rights. To get the rights back if theyll come back should not be a big deal. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:08, 24 April 2026 (UTC)
:: When they don't reply by May 19, feel free (or any custodian) to do so. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:28, 25 April 2026 (UTC)

==Curator inactivity review==
These curators haven't been active for &gt; 2 years. As per the [[Wikiversity:Curatorship|curatorship policy]]:
* [[Special:Log/Cody naccarato]] was notified on their talk page by [[User:Codename Noreste|Codename Noreste]] on 24 Apr 2026
* [[Special:Log/Praxidicae]] was notified on their talk page by [[User:Codename Noreste|Codename Noreste]] on 24 Apr 2026
* [[Special:Log/Tegel]] was notified on their talk page by [[User:Jtneill|Jtneill]] notified their talk page on 16 May 2026

The policy allows a month to hear from these users. If no response, a custodian will remove their curator rights.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:14, 16 May 2026 (UTC)
: For Cody naccarato and Praxidicae, their rights are to be removed by the 19th of May if they don't respond either here or on their talk page. For Tegel, the removal will happen on the 16th of June, probably. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:13, 16 May 2026 (UTC)
::Should be 24 May for Cody naccarato and Praxidicae? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:11, 16 May 2026 (UTC)
::: I made [[#Inactive curators]] on the 19th of April. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:18, 17 May 2026 (UTC)
::::OK, I see (had missed that thread, sorry - I've now moved the the 3 inactivity topics to be adjacent).
::::I'm thinking the curator policy indicates one month from user talk page notification? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:44, 17 May 2026 (UTC)
::::: Yes. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:49, 17 May 2026 (UTC)

== [[Wikiversity:Deletion policy]] proposed as policy ==

[[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]].

This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]].

Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy.

=== Voting ===
*{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:33, 18 May 2026 (UTC)
*{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC)

=== Comments ===</text>
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== Technical Request:  Courtesy link.. ==

[[Template_talk:Information#Background_must_have_color_defined_as_well]] [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
: I can't edit the template directly as it need an sysop/interface admin to do it. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
:: Also if the Template field of - https://en.wikiversity.org/wiki/Special:LintErrors/night-mode-unaware-background-color is examined, there is poential for an admin to clear a substantial proportion of these by implmenting a simmilar fix to the indciated templates (and underlying stylesheets). It would be nice to clear things like Project box and others, as many other templates (and thus pages depend on them.) :)
[[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)

:I think it would be best to grant you interface admin rights for a short period of time to make these changes. However, I still have doubts about the suitability of this solution, which may cause other problems and no one has explained to me why dark mode has to be implemented this way @[[User:ShakespeareFan00|ShakespeareFan00]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:43, 20 March 2026 (UTC)

: I would have reservations about holding such rights, which is why I was trying to do what I could without needing them. However if it is the only way to get the required changes made, I would suggest asking on Wikipedia to find technical editors, willing to undertake the changes needed. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:32, 21 March 2026 (UTC)

== WikiEducator has closed ==

Some of you may know of a similar project to Wikiversity, called [https://wikieducator.org/Main_Page WikiEducator], championed by [https://oerfoundation.org/about/staff/wayne-mackintosh/ Wayne Mackintosh][https://www.linkedin.com/posts/waynemackintosh_important-notice-about-the-oer-foundation-activity-7405113051688931329-Nhm9/][https://openeducation.nz/killed-not-starved/].

It seems [https://openeducation.nz/terminating-oer-foundation their foundation has closed] and they are no longer operating.

They had done quite a bit of outreach (e.g., in the Pacific and Africa) to get educators using wiki.

The WikiEducator content is still available in MediaWiki - and potentially could be imported to Wikiversity ([https://wikieducator.org/WikiEducator:Copyrights CC-BY-SA] is the default license).

The closing of WikiEducator arguably makes the nurturing of Wikiversity even more important.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:09, 1 April 2026 (UTC)

:I was never active there. If anyone has an account or is otherwise in contact, we may want to copy relevant information here or even at [[:outreach:]]. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:46, 1 April 2026 (UTC)

:: I reached out to [[User:Mackiwg~enwikiversity|Wayne]] in January, and he responded briefly but positively (while travelling).  I wrote to the low-traffic wikieducator mailing list today and got a nice [https://groups.google.com/g/wikieducator/c/r_yIyUw6ZIA reply] from [[user:SteveFoerster|Steve Foerster]] who's interested in helping. If we can figure out a migration path it would be great to adopt at least the main namespace pages here.  
:: A few questions that come to mind: 
:: - would people want to create matching user accounts
:: - are there any namespaces (user/talk?) that should not be moved over
:: We could look at how this was done for the [[m:Wikivoyage/Migration]] wikivoyage migration. &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 04:27, 1 May 2026 (UTC)
:::That's fantastic, SJ, that you've reached out and that Wayne, Steve, and Jim are receptive—and that you can help! -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:52, 1 May 2026 (UTC)
::::A matching accound makes sense to give credits to the original authors and keep a clean chain of versions. The initial commit into wikiversity could have a &quot;marker with timestamp&quot; similar to signature with a reference where the content's source or a Web archive. This would allow authors to continue there work on wikiversity if they wish. [[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 06:30, 15 May 2026 (UTC)

== Wikinews is ending ==

Apparently mainly due to low editorial activity, low public interest, but also failure to achieve the goals from the proposal for the creation of the project, the Wikinews project is ending after years of discussions ([[Meta:Proposal for Closing Wikinews|some reading]]).

And I would be interested to see how Wikiversity is doing in the monitored metrics. We probably have more editors than Wikinews had, but what about consumers and achieving the goals? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:14, 1 April 2026 (UTC)

:Wikiversity's biggest issue in recent times was the hosting of low-quality, trash content. Thankfully we've done a great job in removing pseudoscience and other embarrassingly trash content (Wikidebates, for example), but the biggest concern moving forward is proper maintenance IMO. I've caught several pseudoscience pages being created within the last few months that could easily have flown under the radar (ex, [[The Kelemen Dilemma: Causal Collapse and Axiomatic Instability]]), so I'd urge our custodians/curators to be on the lookout for this type of content. Usually an AI-overview can point this type of content out relatively well.
:In terms of visibility, I believe Wikiversity is a high-traffic project. I remember my [[Mathematical Properties]] showing up on the first page of Google when searching up &quot;math properties&quot; for the longest time (and is still showing up in the first page 'till this day!). Besides, Wikinews hosted a lot of short-term content (the nature of news articles), while Wikiversity hosts content that can still be useful a decade later (ex, [[A Reader's Guide to Annotation]]). 
:I think we are on a better path than we were a few months ago, and I do want to thank everyone here who has been helping out with maintaining our website! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:48, 1 April 2026 (UTC)
:For what it's worth, the group that did that study has since disbanded, so no one is monitoring the other sister projects in the same way. Additionally, Wikinews had some catastrophic server issues due to the maintenance of [[:m:Extension:DynamicPageList]] which don't apply here. Your questions are still worth addressing, but I just wanted to cut off any concern at the pass about Wikiversity being in the same precarious situation. Wikiversity is definitely the biggest &quot;lagging behind&quot; or &quot;failure&quot; project now that Wikinews is being shuttered, but I don't see any near- or medium-term pathway to closing Wikiversity. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:46, 2 April 2026 (UTC)

:[[w:en:Wikipedia:Wikipedia Signpost/2026-03-31/News and notes|Entirety of Wikinews to be shut down]] (Wikipedia Signpost) -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:03, 11 April 2026 (UTC)

== Add some user rights to the curator user group? ==

By default, only custodians have the ability to mark new pages as patrolled (&lt;code&gt;patrol&lt;/code&gt;) and have their own page creations automatically marked as patrolled (&lt;code&gt;autopatrol&lt;/code&gt;). I am proposing both of the following:
* Curators can mark new pages as patrolled, helping on reducing the backlog of new, unpatrolled pages.
* New pages made by curators will be automatically marked as patrolled by the MediaWiki software.
Before we implement this, I would suggest implementing a proposed guideline for marking new pages as patrolled for curators and custodians.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:32, 17 April 2026 (UTC)

:Agree, &lt;s&gt;also can we also allow curators to undelete pages since they already have the rights to delete them?&lt;/s&gt; [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:54, 18 April 2026 (UTC)
::I think the requirement that undelete NOT be included came from above (meta / stewards / central office). Having access to the undelete page gives access to information that is restricted by their policies to admins (custodians and bureaucrats). -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 20:12, 18 April 2026 (UTC)
::: [[User:PieWriter|PieWriter]], unless if requests for curator and custodian should be RfA-like processes (that is, including voting and comments), then I have to agree with Dave above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:03, 18 April 2026 (UTC)
::::Oh, I didn’t realise that. Withdrawing my comment.. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:08, 19 April 2026 (UTC)
:{{support}} Seems reasonable and would reduce overhead. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 14:35, 18 April 2026 (UTC)
:'''Agree''', implement it also to [[Wikiversity:Curators]] proposal please. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:11, 18 April 2026 (UTC)
: I went ahead and filed [[phab:T424445]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:39, 26 April 2026 (UTC)
: This was completed on 30 April.
: Perhaps we could benefit from some documentation (e.g., [[Wikiversity:Patrol]] or [[Wikiversity:Patrolling]]?) and updates to the curator, custodian, bureaucrat pages? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:49, 19 May 2026 (UTC)
:: Yes, but I would recommend [[Wikiversity:Patrolling]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 02:26, 19 May 2026 (UTC)

== [[Wikiversity:Curators|Curators and curators policy]] ==
{{archive top|There is strong consensus, so [[Wikiversity:Curators]] is now a policy. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:15, 9 May 2026 (UTC)}}
How does it come, that Wikiversity has curators, but Curators policy is still being proposed? How do the curators exists and act if the policy about them havent been approved yet? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:33, 16 October 2025 (UTC)

:It looks as if it is not just curators. The policy on Bureaucratship is still being proposed as well. See [[Wikiversity:Bureaucratship]]. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 18:33, 27 October 2025 (UTC)

:I think its just the nature of a small WMF sister project in that there are lots of drafts, gaps, and potential improvements. In this case, these community would need to vote on those proposed Wikiversity staff policies if we think they're ready. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:08, 3 December 2025 (UTC)
:What? I thought you were getting it approved, Juandev... :) [[User:I&amp;#39;m Mr. Chris|I&amp;#39;m Mr. Chris]] ([[User talk:I&amp;#39;m Mr. Chris|discuss]] • [[Special:Contributions/I&amp;#39;m Mr. Chris|contribs]]) 14:20, 12 February 2026 (UTC)
::Yeah I think this one is important too and we need to aprove it too @[[User:I'm Mr. Chris|I'm Mr. Chris]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 12 February 2026 (UTC)
:::I thinks its ready to made into a policy, it seems to be complete and informative about what the rights does and how to get it. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:08, 15 February 2026 (UTC)
::::Agree -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:00, 27 March 2026 (UTC)

Let's make this the official discussion about adopting the [[Wikiversity:Curators|curators policy]] policy. Your comments are invited and welcome. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:40, 24 April 2026 (UTC)
: There were two similar Colloquium threads in separate places about the proposed curators policy. So I've moved them to be adjacent. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:42, 1 May 2026 (UTC)
{{archive bottom}}

== Wikiversity:Curators to become a policy ==
{{archive top|There is strong consensus, so [[Wikiversity:Curators]] is now a policy. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:16, 9 May 2026 (UTC)}}
I've looked at the discussions about the Curators policy, I've looked at the practices, and it seems to me that there is no dispute about the wording of the policy, and what's more, the community has been using this proposal as if it were an offical policy for several years. Therefore, I propose that [[Wikiversity:Curators]] become a policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:35, 18 April 2026 (UTC)

:{{support}} —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 18 April 2026 (UTC)
:{{support}} —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:21, 18 April 2026 (UTC)
: {{support|Yes, please}}. Especially after when I and PieWriter proposed above, I agree. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:27, 24 April 2026 (UTC)
:: @[[User:Juandev|Juandev]]; as of now, curators now have the user rights &lt;code&gt;autopatrol&lt;/code&gt; and &lt;code&gt;patrol&lt;/code&gt;. Perhaps we should also include that in [[Wikiversity:Custodianship]]? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 12:07, 30 April 2026 (UTC)
:::You meant [[Wikiversity:Curators]] @[[User:Codename Noreste|Codename Noreste]]? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:15, 5 May 2026 (UTC)
:::: I agree that we must develop what rules curators should follow when marking new pages as patrolled; the same can be added for custodians since they can also mark new pages as patrolled. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:37, 5 May 2026 (UTC)
:::::I see, well I think you can just add this to the policy. It is not major change and it probably reflects actual practice or actual technical possibilities for those flags. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:20, 7 May 2026 (UTC)
:{{support}} -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:42, 1 May 2026 (UTC)
:{{Support}} per nom. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 13:32, 1 May 2026 (UTC)
{{Archive bottom}}

== Is anyone interested in Neurodiversity? ==

Is anyone interested in Neurodiversity? Is there anyone here who is interested for Neurodiversity to be &quot;something more&quot; than it already is? Does anyone here consider Neurodiversity one of the &quot;harder topics&quot; to work on or discuss? Does anyone here have an opinion about the [[Neurodiversity Movement]]? So these questions don't appear like &quot;out of a vacuum&quot; I can tell you a bit about my background:

Many years ago I got a psychiatric diagnosis &quot;Asperger's&quot;. After I stepped out of the office and my Äsperger's was 'concluded', I stepped out into the street and thought my first negative thought(but the positive thought followed after). The thought was about concentration camps in the second world war and that the world seemed to be going into the direction of &quot;labeling others&quot;. I was unsure whether this was &quot;real science&quot; and sort of &quot;challenged myself&quot; to make up my own mind after meeting people that had been given this diagnosis. The more adults with this diagnosis I met the more I started seeing &quot;patterns&quot;.

Was it a coincidence that the first person with Asperger's I met reminded me about my father later after I had plenty of times of experience with interacting with him? None of the people I interacted with online through IRC text chat...I felt I got any clue about how &quot;their brains work&quot;. Only when I met one person from the Asperger's chat community in person we both realized that whatever we experienced was akin to the &quot;chaos theory&quot;. He told me about &quot;chaos theory&quot; while I didn't know even what that term meant but I guess I 'read between the lines'. My question that I linger on still today is &quot;did he understand about me what I think I understood about him?&quot;? That our brains had the same configuration? Most autistic adults who meet other autistic adults usually get disappointed. They think the diagnosis will help them meet somebody like themselves and then they realize the great diversity in the autistic spectrum created by Psychiatry.

I later stopped interacting with autistic communities that much, I felt that it did not benefit me. Also Neurodiversity's &quot;neurotypes&quot; interested me for a while until I realized I had &quot;misunderstood everything&quot; about them and how they are used in the Neurodiversity Movement or &quot;Neurodiversity community&quot; if that even can precisely be defined? I doubt it but if you want to contribute to the [[Neurodiversity Movement]]. My previous attempts failed as I got more and more confused. I think a community project needs a community. With a lack of that I don't think it is worth my time. If any of you would like to work on that project let me know on my talk page.

So I was kinda lost and was talking to my friend and psychologist and I realized if I never talk about my idea to anyone in a &quot;comprehensive way&quot; or show that it matters to me nothing is going to ever happen. So I started talking about my &quot;idea&quot; more. Nobody could understand the &quot;idea&quot; because I had not developed my skills regarding where to start...although the process had already started &quot;automatically&quot; and that's why I often think of &quot;well my brain sort of activated me&quot;. I don't feel like I did have a plan and this idea happened. It happened &quot;by itself&quot;. My brain reacted to what I was seeing in a video or stream.

I value interaction highly in this idea. I think it would be helpful to make a community of people who are not paranoid about stuff that can express itself like &quot;don't analyze me!&quot;, &quot;don't compare me to anyone!&quot;. 

On the contrary, more often than not those adults who were diagnosed were actually openly comparing themselves with each other and I think that is healthy in a &quot;science&quot; way if done the &quot;right way&quot; which probably means &quot;Do no harm&quot;.

I found video material is important but I'm very unsure if uploading own video material to Wikimedia Commons would constitute a &quot;reasonable&quot; use of the resources there. Maybe somebody here needs to ask more questions to me that I should answer before that happens. I also know the '''be bold''' so I could just do what I think might be ok. Though I work better in a group as long as I know what &quot;group configurations&quot; help me. This is in a non-profit way. Since the state supported me this might be a way I am trying to &quot;give back&quot; to the state and &quot;the world&quot;. May seem overly ambitious and crazy but this thing gives me energy. It gives me hope when trying to develop this idea. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:47, 23 April 2026 (UTC)

:Thanks for sharing. There is plenty of room for neurodiversity community learning. However, the challenge I think is that the intersection of those interested in (a) ND, and (b) English Wikiversity might be very small (e.g., 1!) at this point in time.
:But don't give up hope. For example, Wikipedia has many more ND-interested editors; maybe consider reaching out to see who might be interested:
:[[w:Category:Wikipedians interested in neurodiversity]]
:You could also start an equivalent category here:
:[[:Category:Wikiversitarians interested in neurodiversity]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:46, 6 May 2026 (UTC)

== Request for comment (global AI policy) ==

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;A [[:m:Requests for comment/Artificial intelligence policy|request for comment]] is currently being held to decide on a global AI policy. {{int:Feedback-thanks-title}} [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 00:58, 26 April 2026 (UTC)&lt;/bdi&gt;
&lt;!-- Message sent by User:Codename Noreste@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=30424282 --&gt;

== Language learning  ==

toki! I am trying to add or see what the toki pona language learning stuff on here is but I don't see anything that is language learning for anything. [[User:Jan Imon|Jan Imon]] ([[User talk:Jan Imon|discuss]] • [[Special:Contributions/Jan Imon|contribs]]) 23:13, 2 May 2026 (UTC) —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:29, 3 May 2026 (UTC)

:We have language materials ([[:Category:Languages]], [[World Languages]], [[Portal:Foreign Language Learning]], [[Portal:Multilingual Studies]]). They are not as developed as I think we would all like and there's not any coverage of Toki Pona, but in principle, we could and would like that. You can also see [[:b:Subject:Languages]] at our sister project Wikibooks. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:33, 3 May 2026 (UTC)

== Timeline format? ==

I’ve been working on the World War II articles, including the [[World War II/Timeline|timeline]], and is there a specific timeline format that should be used? Right now it’s just a table, and there’s no separation between different periods/phases of the war.

I don’t want to use [[mw:Extension:EasyTimeline]] because this will be displaying dates and not time periods. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 01:35, 4 May 2026 (UTC)

:I dont think we have a policy or guideline, how to format a timeline. But you may try to browes wikiversity by Google if someone was dealing with this in the past somewhow @[[User:PhilDaBirdMan|PhilDaBirdMan]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:23, 5 May 2026 (UTC)
::+1 - there's no specific guideline on how to format a timeline, it's really up to you. In my opinion I think the timeline is good. I'd personally bold the dates just to make it easier to separate it from the event description, but that's my personal 2 cents. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:18, 5 May 2026 (UTC)
:::I’ll probably remove links to the dates/years, they’re just Wikipedia pages that shouldn’t be over linked to. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 00:39, 6 May 2026 (UTC)

== Interface administrator for Codename Noreste ==
{{Archive top|After running for a week, there is clear consensus for [[User:Codename Noreste]] to have Interface admin rights for 120 days; implemented until 10 September, 2026 -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:36, 12 May 2026 (UTC)}}

Hello, everyone. I am requesting interface administrator access on this wiki.

The main reasoning is that I would benefit from having the user right &lt;code&gt;editinterface&lt;/code&gt;, which would allow me to make dark mode changes to pages in the MediaWiki namespace, add &lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;mw-parser-output&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt; to some interface pages using templates, handle interface-protected edit requests, and similar stuff. Additionally, I have some knowledge of CSS, and I would like to assist with modifying CSS pages whenever necessary, such as moving MediaWiki common.css code to TemplateStyles CSS pages.

I am requesting the maximum time that is allowed per the [[Wikiversity:Interface administrators|policy]], and I have 2FA enabled on my account. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:55, 6 May 2026 (UTC)

*{{support}} Globally trusted user. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:07, 6 May 2026 (UTC)
*{{support}} Trusted and knowledgeable. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:35, 6 May 2026 (UTC)
*{{support}} WV would benefit from this. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 08:32, 6 May 2026 (UTC)
*{{support}} --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:13, 7 May 2026 (UTC)

:{{Comment}} Could @[[User:Codename Noreste|Codename Noreste]] delete [[MediaWiki:Gadget-WikiSign.js]], which was requested to be deleted @[[User:Koavf|Justin]], @[[User:Jtneill|Jtneill]], @[[User:Atcovi|Atcovi]]? I dont think we need it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:40, 9 May 2026 (UTC)
::Yes - clearly no longer used -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:18, 9 May 2026 (UTC)
::: I can't delete it because I don't have the required permissions to do so.
::: On a side note, if this project has a need for permanent interface administrators, I would suggest that we have a minimum of two IAs, similar to how there must be two CUs and/or suppressors (or none). Maybe Koavf can be a good candidate if I am elected for permanent interface adminship, and I believe that permission shouldn't be removed from someone's own account. Instead, a bureaucrat should do it. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:20, 9 May 2026 (UTC)
::::I am willing and happy to do it, unfortunately, we do not have an appetite for indef IAs and just had a discussion that resulted in a [https://en.wikiversity.org/w/index.php?title=Wikiversity:Interface_administrators&amp;diff=prev&amp;oldid=2807543 consensus that we can have IAs that have the user rights for 14 to 120 days]. So once you have the rights, please make sure to gopher it. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:54, 9 May 2026 (UTC)
:::::@[[User:Koavf|Koavf]] give it time. Look at me, I was in favor of shorter time, now I am looking back to times, when custodians could do it without the need of extra flag. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:31, 9 May 2026 (UTC)
::::::Here's hoping. I think it would reduce administrative overhead, but that's just me and I'm not a bureaucrat here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:33, 9 May 2026 (UTC)
::::Complicated. Where are the times, admins could do everything! [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:27, 9 May 2026 (UTC)
{{archive bottom}}

== [[Wikiversity:Bureaucratship]] to become a policy ==
{{archive top|'''Approved - now a policy'''. 5 supports + 1 nominator. No objections.}}
Following the recent approval of [[Wikiversity:Curators]] as a policy, I think [[Wikiversity:Bureaucratship]] may also be ready for policy status.

Please share your views about whether bureaucratship is ready to become a policy, or whether further revisions are needed.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 13:58, 9 May 2026 (UTC)
: I added a logo about that user group, but other than that, it looks good to me. {{support}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:38, 9 May 2026 (UTC)
:I think that the consensus on this policy is proven by years of using it without further changes. But I I have to say weather I agree with this to become a policy, than of course {{support}}. It works and there were no major issues with it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:45, 9 May 2026 (UTC)
:{{support}} no issues. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:51, 10 May 2026 (UTC)
:{{support}} [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:37, 11 May 2026 (UTC)
:{{support}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:51, 12 May 2026 (UTC)
{{archive bottom}}

== Reminder about custodian-related pages ==

I would like to remind the community about what the following custodian pages are:
* [[Wikiversity:Request custodian action]] is for requesting actions to be done by custodians, and
* [[Wikiversity:Notices for custodians]] is for notices of interest to custodians, like an administrator's noticeboard
Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:12, 12 May 2026 (UTC)

:Thanks - I needed this reminder :) -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:21, 12 May 2026 (UTC)

== [[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/&amp;#126;2026-28640-56|&amp;#126;2026-28640-56]] ([[User talk:&amp;#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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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)

== Create a pseudo-bot user group? ==

I would like to propose adding a new user group to Wikiversity: Pseudo-bot (&lt;code&gt;flood&lt;/code&gt;). This will allow users to perform repetitive actions without flushing the recent changes feed (with only the &lt;code&gt;bot&lt;/code&gt; user right). However, I would suggest that for the pseudo-bot user group:
* It can be granted and revoked by custodians. &lt;s&gt;However, can curators add and remove pseudo-bot from their own accounts (and not others)?&lt;/s&gt;
* Users can remove themselves from it.
* A guideline might be necessary about the information and usage of it.
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:31, 14 May 2026 (UTC)

:This sounds good. Which other wiki could we model this user group on? e.g., [[b:Wikibooks:Pseudo-bots]]? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:19, 15 May 2026 (UTC)
::@[[User:Jtneill|Jtneill]] Wikiquote has a similar group: [[:wikiquote:Special:ListGroupRights]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:25, 15 May 2026 (UTC)
: Should we allow curators to add and remove themselves from the pseudobot user group (from their own account) as well? I see no objections to creating the user group. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:20, 18 May 2026 (UTC)
::My thinking is perhaps not curators by default because there should be clear visibility about their actions until they are well trusted. Let's draft a guideline or proposed policy ([[Wikiversity:Pseudo-bots]]) for the proposed user group. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:39, 18 May 2026 (UTC)
::: A solution is that they can ask any custodian to grant that group, and to remove themselves when done. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:17, 19 May 2026 (UTC)
:::: Yes, that sounds good. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:12, 19 May 2026 (UTC)

== Coming over From wikinews ==
Any chance someone could help me if you are allowed to write news articles here since wikinews is going read only mode soon, thank you! [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:43, 1 May 2026 (UTC)

:The scope of Wikiversity is very broad and is basically about more-or-less any learning material. We have made it a point to not have duplicative content of other WMF projects, but since Wikinews is being shuttered, I personally am fine with writing news articles here. One thing that is not controversial at all is a learning resource &lt;em&gt;about&lt;/em&gt; how to write news: that could be hugely useful here and could involve the process of writing news stories to learn and to share back and forth with an editor or fact-checker. In fact, I'd support an entire namespace dedicated to keeping the notion of Wikinews alive here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:38, 1 May 2026 (UTC)
::Thank you so much! How do I start? Cheers! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:07, 2 May 2026 (UTC)
:::I think it's premature to start just making news articles en masse, but if you want to start discussing the topic of citizen journalism, you can do that now. [[:Category:Journalism]] already has some material, so you can start by seeing what we already have, how you can refine that, etc. You can definitely have learning resources with collaborators who want to learn about journalism ASAP. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:24, 2 May 2026 (UTC)
::::thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:38, 2 May 2026 (UTC)
::::If I could try and start one News Article could you please tell me how to go about it? Like what style of writing like Wikinews or something else? Thank you Justin! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:48, 2 May 2026 (UTC)
:::::Honestly, there are very few policies and guidelines here. I think the best way to write a news story would be in a manner that is obvious and instructive. So, for instance, it's common to use the &quot;pyramid style&quot; when you're writing news, so if you were to write a story that makes it very clear that you are using that approach, that would be helpful. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 02:08, 2 May 2026 (UTC)
::::::cool thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 02:13, 2 May 2026 (UTC)
::::::im ready to write @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 21:30, 13 May 2026 (UTC)
:::::::I think we should get more local consensus for a big project like including the entirety of the scope of Wikinews here. Again, I support it personally. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:55, 13 May 2026 (UTC)
::::::::ok lets begin. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:15, 13 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 &quot;practicing journalism&quot; 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)

===Comments and questions===
:Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice.
:Initial questions:
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
:* What are &quot;active editions&quot;?
:* 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 &quot;active editions&quot;?
::*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 &lt;em&gt;about&lt;/em&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)

== Inactivity policy for Curators ==

I was wondering if there is a specific inactivity polity for curators (semi-admins) as I am pretty sure the global policy does not apply to them as they are not ''fully'' sysops. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:20, 15 February 2026 (UTC)

:Unfortunately, I don't see an inactivity policy, but if we were to create such a new policy for curators, it should be the same for custodians (administrators). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:45, 15 February 2026 (UTC)
::@[[User:Codename Noreste|Codename Noreste]] There is currently none, that I could find, for custodians either. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:47, 17 February 2026 (UTC)
:::I think we should propose a local inactivity policy for custodians (and by extension, curators), which should be at least one year without any edits ''and'' logged actions. However, I don't know which page should it be when the inactivity removal procedure starts. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:53, 17 February 2026 (UTC)
::::@[[User:Codename Noreste|Codename Noreste]] In theory, there should be a section added at [[WV:Candidates for custodianship]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:55, 17 February 2026 (UTC)
::::: To be consistent with the [[meta:Admin activity review|global period of 2 years inactivity]] for en.wv  [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship#How are bureaucrats removed?|Bureaucrats]] we could add something like this to [[Wikiversity:Curators]]:
::::::The maximum time period of inactivity &lt;u&gt;without community review&lt;/u&gt; for curators is two years (consistent with  the [[:meta:Category:Global policies|global policy]] described at [[meta:Admin activity review|Admin activity review]] which applies for [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship|Bureaucrats]]). After that time a custodian will remove the rights.
::::: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:51, 27 March 2026 (UTC)
:::::Yup, I agree with Jtneill, there is a policy proposal for Wikiversity:Curators, where it should be logically deployed. The question is if we are ready to aprove the policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:43, 17 April 2026 (UTC)
:::::: I agree, but we should notify the colloquium about inactive curators, just like a steward would do for inactive custodians and bureaucrats per [[:m:Admin activity review|AAR]]. What is the minimum timeframe an inactive curator should receive so they can respond they would keep their rights? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:49, 17 April 2026 (UTC)
:I incorporated these suggestions into the proposed curators policy. Please review/comment/improve. Summary: 2 years, notify curator's user page, then remove rights after 1 month: [[Wikiversity:Curators#Inactivity]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:59, 24 April 2026 (UTC)
:: @[[User:Jtneill|Jtneill]] I created [[Template:Inactive curator]] for this. Feel free to make any changes or improvements. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:29, 24 April 2026 (UTC)
:::Wondering, should we also have:
:::* {{tl|Inactive custodian}}
:::* {{tl|Inactive bureaucrat}}
:::or perhaps just a single template with a parameter(s) for the user right(s)/role(s)? e.g., 
:::* if a custodian is inactive for 2 years, then custodian and curator rights are to be removed and 
:::* if a bureaucrat is inactive for 2 years, then bureaucrat, custodian, and curator rights are to to be removed.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:58, 13 May 2026 (UTC)
:::: I would probably modify that template when we actually develop our own inactivity policy, because we're currently under the AAR (a steward notifies the colloquium with [[m:Admin activity review/Notice to communities]], and inactive advanced right holders with [[m:Admin activity review/Notice to inactive right holders]]). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:16, 13 May 2026 (UTC)
:::::Ah, I see. Yes, that makes sense. Thankyou. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:21, 15 May 2026 (UTC)
: In that case, should we develop our own inactivity policy (e.g. on [[Wikiversity:Inactivity policy]] or [[Wikiversity:Support staff/Inactivity]])? I would list the general inactivity part, the process, etc. Once it's approved as a policy, I will [[m:Stewards' noticeboard|notify the stewards]]. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:30, 16 May 2026 (UTC)
::Originally, I would have thought that, for a small wiki like en.wv, it made sense to leave inactivity monitoring to the stewards. However, with the creation of the curator user group, we have already taken on local responsibility for monitoring inactivity in at least one advanced-rights group. Extending this to custodians and bureaucrats would not add much additional overhead and would provide a more consistent and transparent local administrative process.
::One option would be to develop a single, centralised policy covering all advanced-rights groups.
::An alternative would be to include an ==Inactivity== section on each relevant policy page (e.g., we already have [[Wikiversity:Curatorship#Inactivity]], but not yet in the custodianship, and bureaucratship policy pages). This approach would allow some flexibility because different user groups may warrant different criteria (such as inactivity thresholds, qualifying activity, or review procedures).
::A hybrid approach may be best: maintain separate inactivity sections within each user-group policy page, while transcluding these into a central overview page such as Codename Noreste suggests. This would preserve clarity at the local policy level while also providing a single reference point for consistency and oversight. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:09, 16 May 2026 (UTC)
::: I would suggest we develop a centralized inactivity policy page, and include a short summarized section of that page, on the support staff user group pages. We must also include a link to that policy page if we were to add &lt;nowiki&gt;== Inactivity ==&lt;/nowiki&gt; to each of those user group pages. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:48, 17 May 2026 (UTC)

== Inactive curators ==

Hello, even though [[Wikiversity:Curators]] is not a policy yet, there are curators listed here that have been inactive for two years or more:
* {{user|Cody naccarato}} (last edit on 13 Dec 2022, last logged action on 10 Dec 2022)
* {{user|Praxidicae}} (last edit on 10 Sep 2022, last logged action on 12 Sep 2022)
[[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 19 April 2026 (UTC)

:Yup, I would remove the rights. To get the rights back if theyll come back should not be a big deal. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:08, 24 April 2026 (UTC)
:: When they don't reply by May 19, feel free (or any custodian) to do so. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:28, 25 April 2026 (UTC)

==Curator inactivity review==
These curators haven't been active for &gt; 2 years. As per the [[Wikiversity:Curatorship|curatorship policy]]:
* [[Special:Log/Cody naccarato]] was notified on their talk page by [[User:Codename Noreste|Codename Noreste]] on 24 Apr 2026
* [[Special:Log/Praxidicae]] was notified on their talk page by [[User:Codename Noreste|Codename Noreste]] on 24 Apr 2026
* [[Special:Log/Tegel]] was notified on their talk page by [[User:Jtneill|Jtneill]] notified their talk page on 16 May 2026

The policy allows a month to hear from these users. If no response, a custodian will remove their curator rights.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:14, 16 May 2026 (UTC)
: For Cody naccarato and Praxidicae, their rights are to be removed by the 19th of May if they don't respond either here or on their talk page. For Tegel, the removal will happen on the 16th of June, probably. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:13, 16 May 2026 (UTC)
::Should be 24 May for Cody naccarato and Praxidicae? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:11, 16 May 2026 (UTC)
::: I made [[#Inactive curators]] on the 19th of April. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:18, 17 May 2026 (UTC)
::::OK, I see (had missed that thread, sorry - I've now moved the the 3 inactivity topics to be adjacent).
::::I'm thinking the curator policy indicates one month from user talk page notification? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:44, 17 May 2026 (UTC)
::::: Yes. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:49, 17 May 2026 (UTC)
: @[[User:Juandev|Juandev]] and @[[User:Jtneill|Jtneill]]: feel free to remove Cody naccarato and Praxidicae's curator permissions. They have not responded at all after one month. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:29, 19 May 2026 (UTC)

== [[Wikiversity:Deletion policy]] proposed as policy ==

[[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]].

This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]].

Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy.

=== Voting ===
*{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:33, 18 May 2026 (UTC)
*{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC)

=== Comments ===</text>
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== Technical Request:  Courtesy link.. ==

[[Template_talk:Information#Background_must_have_color_defined_as_well]] [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
: I can't edit the template directly as it need an sysop/interface admin to do it. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
:: Also if the Template field of - https://en.wikiversity.org/wiki/Special:LintErrors/night-mode-unaware-background-color is examined, there is poential for an admin to clear a substantial proportion of these by implmenting a simmilar fix to the indciated templates (and underlying stylesheets). It would be nice to clear things like Project box and others, as many other templates (and thus pages depend on them.) :)
[[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)

:I think it would be best to grant you interface admin rights for a short period of time to make these changes. However, I still have doubts about the suitability of this solution, which may cause other problems and no one has explained to me why dark mode has to be implemented this way @[[User:ShakespeareFan00|ShakespeareFan00]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:43, 20 March 2026 (UTC)

: I would have reservations about holding such rights, which is why I was trying to do what I could without needing them. However if it is the only way to get the required changes made, I would suggest asking on Wikipedia to find technical editors, willing to undertake the changes needed. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:32, 21 March 2026 (UTC)

== WikiEducator has closed ==

Some of you may know of a similar project to Wikiversity, called [https://wikieducator.org/Main_Page WikiEducator], championed by [https://oerfoundation.org/about/staff/wayne-mackintosh/ Wayne Mackintosh][https://www.linkedin.com/posts/waynemackintosh_important-notice-about-the-oer-foundation-activity-7405113051688931329-Nhm9/][https://openeducation.nz/killed-not-starved/].

It seems [https://openeducation.nz/terminating-oer-foundation their foundation has closed] and they are no longer operating.

They had done quite a bit of outreach (e.g., in the Pacific and Africa) to get educators using wiki.

The WikiEducator content is still available in MediaWiki - and potentially could be imported to Wikiversity ([https://wikieducator.org/WikiEducator:Copyrights CC-BY-SA] is the default license).

The closing of WikiEducator arguably makes the nurturing of Wikiversity even more important.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:09, 1 April 2026 (UTC)

:I was never active there. If anyone has an account or is otherwise in contact, we may want to copy relevant information here or even at [[:outreach:]]. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:46, 1 April 2026 (UTC)

:: I reached out to [[User:Mackiwg~enwikiversity|Wayne]] in January, and he responded briefly but positively (while travelling).  I wrote to the low-traffic wikieducator mailing list today and got a nice [https://groups.google.com/g/wikieducator/c/r_yIyUw6ZIA reply] from [[user:SteveFoerster|Steve Foerster]] who's interested in helping. If we can figure out a migration path it would be great to adopt at least the main namespace pages here.  
:: A few questions that come to mind: 
:: - would people want to create matching user accounts
:: - are there any namespaces (user/talk?) that should not be moved over
:: We could look at how this was done for the [[m:Wikivoyage/Migration]] wikivoyage migration. &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 04:27, 1 May 2026 (UTC)
:::That's fantastic, SJ, that you've reached out and that Wayne, Steve, and Jim are receptive—and that you can help! -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:52, 1 May 2026 (UTC)
::::A matching accound makes sense to give credits to the original authors and keep a clean chain of versions. The initial commit into wikiversity could have a &quot;marker with timestamp&quot; similar to signature with a reference where the content's source or a Web archive. This would allow authors to continue there work on wikiversity if they wish. [[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 06:30, 15 May 2026 (UTC)

== Wikinews is ending ==

Apparently mainly due to low editorial activity, low public interest, but also failure to achieve the goals from the proposal for the creation of the project, the Wikinews project is ending after years of discussions ([[Meta:Proposal for Closing Wikinews|some reading]]).

And I would be interested to see how Wikiversity is doing in the monitored metrics. We probably have more editors than Wikinews had, but what about consumers and achieving the goals? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:14, 1 April 2026 (UTC)

:Wikiversity's biggest issue in recent times was the hosting of low-quality, trash content. Thankfully we've done a great job in removing pseudoscience and other embarrassingly trash content (Wikidebates, for example), but the biggest concern moving forward is proper maintenance IMO. I've caught several pseudoscience pages being created within the last few months that could easily have flown under the radar (ex, [[The Kelemen Dilemma: Causal Collapse and Axiomatic Instability]]), so I'd urge our custodians/curators to be on the lookout for this type of content. Usually an AI-overview can point this type of content out relatively well.
:In terms of visibility, I believe Wikiversity is a high-traffic project. I remember my [[Mathematical Properties]] showing up on the first page of Google when searching up &quot;math properties&quot; for the longest time (and is still showing up in the first page 'till this day!). Besides, Wikinews hosted a lot of short-term content (the nature of news articles), while Wikiversity hosts content that can still be useful a decade later (ex, [[A Reader's Guide to Annotation]]). 
:I think we are on a better path than we were a few months ago, and I do want to thank everyone here who has been helping out with maintaining our website! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:48, 1 April 2026 (UTC)
:For what it's worth, the group that did that study has since disbanded, so no one is monitoring the other sister projects in the same way. Additionally, Wikinews had some catastrophic server issues due to the maintenance of [[:m:Extension:DynamicPageList]] which don't apply here. Your questions are still worth addressing, but I just wanted to cut off any concern at the pass about Wikiversity being in the same precarious situation. Wikiversity is definitely the biggest &quot;lagging behind&quot; or &quot;failure&quot; project now that Wikinews is being shuttered, but I don't see any near- or medium-term pathway to closing Wikiversity. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:46, 2 April 2026 (UTC)

:[[w:en:Wikipedia:Wikipedia Signpost/2026-03-31/News and notes|Entirety of Wikinews to be shut down]] (Wikipedia Signpost) -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:03, 11 April 2026 (UTC)

== Add some user rights to the curator user group? ==

By default, only custodians have the ability to mark new pages as patrolled (&lt;code&gt;patrol&lt;/code&gt;) and have their own page creations automatically marked as patrolled (&lt;code&gt;autopatrol&lt;/code&gt;). I am proposing both of the following:
* Curators can mark new pages as patrolled, helping on reducing the backlog of new, unpatrolled pages.
* New pages made by curators will be automatically marked as patrolled by the MediaWiki software.
Before we implement this, I would suggest implementing a proposed guideline for marking new pages as patrolled for curators and custodians.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:32, 17 April 2026 (UTC)

:Agree, &lt;s&gt;also can we also allow curators to undelete pages since they already have the rights to delete them?&lt;/s&gt; [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:54, 18 April 2026 (UTC)
::I think the requirement that undelete NOT be included came from above (meta / stewards / central office). Having access to the undelete page gives access to information that is restricted by their policies to admins (custodians and bureaucrats). -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 20:12, 18 April 2026 (UTC)
::: [[User:PieWriter|PieWriter]], unless if requests for curator and custodian should be RfA-like processes (that is, including voting and comments), then I have to agree with Dave above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:03, 18 April 2026 (UTC)
::::Oh, I didn’t realise that. Withdrawing my comment.. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:08, 19 April 2026 (UTC)
:{{support}} Seems reasonable and would reduce overhead. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 14:35, 18 April 2026 (UTC)
:'''Agree''', implement it also to [[Wikiversity:Curators]] proposal please. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:11, 18 April 2026 (UTC)
: I went ahead and filed [[phab:T424445]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:39, 26 April 2026 (UTC)
: This was completed on 30 April.
: Perhaps we could benefit from some documentation (e.g., [[Wikiversity:Patrol]] or [[Wikiversity:Patrolling]]?) and updates to the curator, custodian, bureaucrat pages? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:49, 19 May 2026 (UTC)
:: Yes, but I would recommend [[Wikiversity:Patrolling]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 02:26, 19 May 2026 (UTC)

== [[Wikiversity:Curators|Curators and curators policy]] ==
{{archive top|There is strong consensus, so [[Wikiversity:Curators]] is now a policy. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:15, 9 May 2026 (UTC)}}
How does it come, that Wikiversity has curators, but Curators policy is still being proposed? How do the curators exists and act if the policy about them havent been approved yet? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:33, 16 October 2025 (UTC)

:It looks as if it is not just curators. The policy on Bureaucratship is still being proposed as well. See [[Wikiversity:Bureaucratship]]. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 18:33, 27 October 2025 (UTC)

:I think its just the nature of a small WMF sister project in that there are lots of drafts, gaps, and potential improvements. In this case, these community would need to vote on those proposed Wikiversity staff policies if we think they're ready. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:08, 3 December 2025 (UTC)
:What? I thought you were getting it approved, Juandev... :) [[User:I&amp;#39;m Mr. Chris|I&amp;#39;m Mr. Chris]] ([[User talk:I&amp;#39;m Mr. Chris|discuss]] • [[Special:Contributions/I&amp;#39;m Mr. Chris|contribs]]) 14:20, 12 February 2026 (UTC)
::Yeah I think this one is important too and we need to aprove it too @[[User:I'm Mr. Chris|I'm Mr. Chris]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 12 February 2026 (UTC)
:::I thinks its ready to made into a policy, it seems to be complete and informative about what the rights does and how to get it. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:08, 15 February 2026 (UTC)
::::Agree -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:00, 27 March 2026 (UTC)

Let's make this the official discussion about adopting the [[Wikiversity:Curators|curators policy]] policy. Your comments are invited and welcome. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:40, 24 April 2026 (UTC)
: There were two similar Colloquium threads in separate places about the proposed curators policy. So I've moved them to be adjacent. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:42, 1 May 2026 (UTC)
{{archive bottom}}

== Wikiversity:Curators to become a policy ==
{{archive top|There is strong consensus, so [[Wikiversity:Curators]] is now a policy. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:16, 9 May 2026 (UTC)}}
I've looked at the discussions about the Curators policy, I've looked at the practices, and it seems to me that there is no dispute about the wording of the policy, and what's more, the community has been using this proposal as if it were an offical policy for several years. Therefore, I propose that [[Wikiversity:Curators]] become a policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:35, 18 April 2026 (UTC)

:{{support}} —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 18 April 2026 (UTC)
:{{support}} —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:21, 18 April 2026 (UTC)
: {{support|Yes, please}}. Especially after when I and PieWriter proposed above, I agree. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:27, 24 April 2026 (UTC)
:: @[[User:Juandev|Juandev]]; as of now, curators now have the user rights &lt;code&gt;autopatrol&lt;/code&gt; and &lt;code&gt;patrol&lt;/code&gt;. Perhaps we should also include that in [[Wikiversity:Custodianship]]? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 12:07, 30 April 2026 (UTC)
:::You meant [[Wikiversity:Curators]] @[[User:Codename Noreste|Codename Noreste]]? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:15, 5 May 2026 (UTC)
:::: I agree that we must develop what rules curators should follow when marking new pages as patrolled; the same can be added for custodians since they can also mark new pages as patrolled. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:37, 5 May 2026 (UTC)
:::::I see, well I think you can just add this to the policy. It is not major change and it probably reflects actual practice or actual technical possibilities for those flags. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:20, 7 May 2026 (UTC)
:{{support}} -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:42, 1 May 2026 (UTC)
:{{Support}} per nom. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 13:32, 1 May 2026 (UTC)
{{Archive bottom}}

== Is anyone interested in Neurodiversity? ==

Is anyone interested in Neurodiversity? Is there anyone here who is interested for Neurodiversity to be &quot;something more&quot; than it already is? Does anyone here consider Neurodiversity one of the &quot;harder topics&quot; to work on or discuss? Does anyone here have an opinion about the [[Neurodiversity Movement]]? So these questions don't appear like &quot;out of a vacuum&quot; I can tell you a bit about my background:

Many years ago I got a psychiatric diagnosis &quot;Asperger's&quot;. After I stepped out of the office and my Äsperger's was 'concluded', I stepped out into the street and thought my first negative thought(but the positive thought followed after). The thought was about concentration camps in the second world war and that the world seemed to be going into the direction of &quot;labeling others&quot;. I was unsure whether this was &quot;real science&quot; and sort of &quot;challenged myself&quot; to make up my own mind after meeting people that had been given this diagnosis. The more adults with this diagnosis I met the more I started seeing &quot;patterns&quot;.

Was it a coincidence that the first person with Asperger's I met reminded me about my father later after I had plenty of times of experience with interacting with him? None of the people I interacted with online through IRC text chat...I felt I got any clue about how &quot;their brains work&quot;. Only when I met one person from the Asperger's chat community in person we both realized that whatever we experienced was akin to the &quot;chaos theory&quot;. He told me about &quot;chaos theory&quot; while I didn't know even what that term meant but I guess I 'read between the lines'. My question that I linger on still today is &quot;did he understand about me what I think I understood about him?&quot;? That our brains had the same configuration? Most autistic adults who meet other autistic adults usually get disappointed. They think the diagnosis will help them meet somebody like themselves and then they realize the great diversity in the autistic spectrum created by Psychiatry.

I later stopped interacting with autistic communities that much, I felt that it did not benefit me. Also Neurodiversity's &quot;neurotypes&quot; interested me for a while until I realized I had &quot;misunderstood everything&quot; about them and how they are used in the Neurodiversity Movement or &quot;Neurodiversity community&quot; if that even can precisely be defined? I doubt it but if you want to contribute to the [[Neurodiversity Movement]]. My previous attempts failed as I got more and more confused. I think a community project needs a community. With a lack of that I don't think it is worth my time. If any of you would like to work on that project let me know on my talk page.

So I was kinda lost and was talking to my friend and psychologist and I realized if I never talk about my idea to anyone in a &quot;comprehensive way&quot; or show that it matters to me nothing is going to ever happen. So I started talking about my &quot;idea&quot; more. Nobody could understand the &quot;idea&quot; because I had not developed my skills regarding where to start...although the process had already started &quot;automatically&quot; and that's why I often think of &quot;well my brain sort of activated me&quot;. I don't feel like I did have a plan and this idea happened. It happened &quot;by itself&quot;. My brain reacted to what I was seeing in a video or stream.

I value interaction highly in this idea. I think it would be helpful to make a community of people who are not paranoid about stuff that can express itself like &quot;don't analyze me!&quot;, &quot;don't compare me to anyone!&quot;. 

On the contrary, more often than not those adults who were diagnosed were actually openly comparing themselves with each other and I think that is healthy in a &quot;science&quot; way if done the &quot;right way&quot; which probably means &quot;Do no harm&quot;.

I found video material is important but I'm very unsure if uploading own video material to Wikimedia Commons would constitute a &quot;reasonable&quot; use of the resources there. Maybe somebody here needs to ask more questions to me that I should answer before that happens. I also know the '''be bold''' so I could just do what I think might be ok. Though I work better in a group as long as I know what &quot;group configurations&quot; help me. This is in a non-profit way. Since the state supported me this might be a way I am trying to &quot;give back&quot; to the state and &quot;the world&quot;. May seem overly ambitious and crazy but this thing gives me energy. It gives me hope when trying to develop this idea. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:47, 23 April 2026 (UTC)

:Thanks for sharing. There is plenty of room for neurodiversity community learning. However, the challenge I think is that the intersection of those interested in (a) ND, and (b) English Wikiversity might be very small (e.g., 1!) at this point in time.
:But don't give up hope. For example, Wikipedia has many more ND-interested editors; maybe consider reaching out to see who might be interested:
:[[w:Category:Wikipedians interested in neurodiversity]]
:You could also start an equivalent category here:
:[[:Category:Wikiversitarians interested in neurodiversity]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:46, 6 May 2026 (UTC)

== Request for comment (global AI policy) ==

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;A [[:m:Requests for comment/Artificial intelligence policy|request for comment]] is currently being held to decide on a global AI policy. {{int:Feedback-thanks-title}} [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 00:58, 26 April 2026 (UTC)&lt;/bdi&gt;
&lt;!-- Message sent by User:Codename Noreste@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=30424282 --&gt;

== Language learning  ==

toki! I am trying to add or see what the toki pona language learning stuff on here is but I don't see anything that is language learning for anything. [[User:Jan Imon|Jan Imon]] ([[User talk:Jan Imon|discuss]] • [[Special:Contributions/Jan Imon|contribs]]) 23:13, 2 May 2026 (UTC) —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:29, 3 May 2026 (UTC)

:We have language materials ([[:Category:Languages]], [[World Languages]], [[Portal:Foreign Language Learning]], [[Portal:Multilingual Studies]]). They are not as developed as I think we would all like and there's not any coverage of Toki Pona, but in principle, we could and would like that. You can also see [[:b:Subject:Languages]] at our sister project Wikibooks. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:33, 3 May 2026 (UTC)

== Timeline format? ==

I’ve been working on the World War II articles, including the [[World War II/Timeline|timeline]], and is there a specific timeline format that should be used? Right now it’s just a table, and there’s no separation between different periods/phases of the war.

I don’t want to use [[mw:Extension:EasyTimeline]] because this will be displaying dates and not time periods. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 01:35, 4 May 2026 (UTC)

:I dont think we have a policy or guideline, how to format a timeline. But you may try to browes wikiversity by Google if someone was dealing with this in the past somewhow @[[User:PhilDaBirdMan|PhilDaBirdMan]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:23, 5 May 2026 (UTC)
::+1 - there's no specific guideline on how to format a timeline, it's really up to you. In my opinion I think the timeline is good. I'd personally bold the dates just to make it easier to separate it from the event description, but that's my personal 2 cents. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:18, 5 May 2026 (UTC)
:::I’ll probably remove links to the dates/years, they’re just Wikipedia pages that shouldn’t be over linked to. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 00:39, 6 May 2026 (UTC)

== Interface administrator for Codename Noreste ==
{{Archive top|After running for a week, there is clear consensus for [[User:Codename Noreste]] to have Interface admin rights for 120 days; implemented until 10 September, 2026 -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:36, 12 May 2026 (UTC)}}

Hello, everyone. I am requesting interface administrator access on this wiki.

The main reasoning is that I would benefit from having the user right &lt;code&gt;editinterface&lt;/code&gt;, which would allow me to make dark mode changes to pages in the MediaWiki namespace, add &lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;mw-parser-output&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt; to some interface pages using templates, handle interface-protected edit requests, and similar stuff. Additionally, I have some knowledge of CSS, and I would like to assist with modifying CSS pages whenever necessary, such as moving MediaWiki common.css code to TemplateStyles CSS pages.

I am requesting the maximum time that is allowed per the [[Wikiversity:Interface administrators|policy]], and I have 2FA enabled on my account. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:55, 6 May 2026 (UTC)

*{{support}} Globally trusted user. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:07, 6 May 2026 (UTC)
*{{support}} Trusted and knowledgeable. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:35, 6 May 2026 (UTC)
*{{support}} WV would benefit from this. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 08:32, 6 May 2026 (UTC)
*{{support}} --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:13, 7 May 2026 (UTC)

:{{Comment}} Could @[[User:Codename Noreste|Codename Noreste]] delete [[MediaWiki:Gadget-WikiSign.js]], which was requested to be deleted @[[User:Koavf|Justin]], @[[User:Jtneill|Jtneill]], @[[User:Atcovi|Atcovi]]? I dont think we need it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:40, 9 May 2026 (UTC)
::Yes - clearly no longer used -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:18, 9 May 2026 (UTC)
::: I can't delete it because I don't have the required permissions to do so.
::: On a side note, if this project has a need for permanent interface administrators, I would suggest that we have a minimum of two IAs, similar to how there must be two CUs and/or suppressors (or none). Maybe Koavf can be a good candidate if I am elected for permanent interface adminship, and I believe that permission shouldn't be removed from someone's own account. Instead, a bureaucrat should do it. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:20, 9 May 2026 (UTC)
::::I am willing and happy to do it, unfortunately, we do not have an appetite for indef IAs and just had a discussion that resulted in a [https://en.wikiversity.org/w/index.php?title=Wikiversity:Interface_administrators&amp;diff=prev&amp;oldid=2807543 consensus that we can have IAs that have the user rights for 14 to 120 days]. So once you have the rights, please make sure to gopher it. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:54, 9 May 2026 (UTC)
:::::@[[User:Koavf|Koavf]] give it time. Look at me, I was in favor of shorter time, now I am looking back to times, when custodians could do it without the need of extra flag. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:31, 9 May 2026 (UTC)
::::::Here's hoping. I think it would reduce administrative overhead, but that's just me and I'm not a bureaucrat here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:33, 9 May 2026 (UTC)
::::Complicated. Where are the times, admins could do everything! [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:27, 9 May 2026 (UTC)
{{archive bottom}}

== [[Wikiversity:Bureaucratship]] to become a policy ==
{{archive top|'''Approved - now a policy'''. 5 supports + 1 nominator. No objections.}}
Following the recent approval of [[Wikiversity:Curators]] as a policy, I think [[Wikiversity:Bureaucratship]] may also be ready for policy status.

Please share your views about whether bureaucratship is ready to become a policy, or whether further revisions are needed.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 13:58, 9 May 2026 (UTC)
: I added a logo about that user group, but other than that, it looks good to me. {{support}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:38, 9 May 2026 (UTC)
:I think that the consensus on this policy is proven by years of using it without further changes. But I I have to say weather I agree with this to become a policy, than of course {{support}}. It works and there were no major issues with it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:45, 9 May 2026 (UTC)
:{{support}} no issues. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:51, 10 May 2026 (UTC)
:{{support}} [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:37, 11 May 2026 (UTC)
:{{support}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:51, 12 May 2026 (UTC)
{{archive bottom}}

== Reminder about custodian-related pages ==

I would like to remind the community about what the following custodian pages are:
* [[Wikiversity:Request custodian action]] is for requesting actions to be done by custodians, and
* [[Wikiversity:Notices for custodians]] is for notices of interest to custodians, like an administrator's noticeboard
Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:12, 12 May 2026 (UTC)

:Thanks - I needed this reminder :) -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:21, 12 May 2026 (UTC)

== [[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/&amp;#126;2026-28640-56|&amp;#126;2026-28640-56]] ([[User talk:&amp;#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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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)

== Create a pseudo-bot user group? ==

I would like to propose adding a new user group to Wikiversity: Pseudo-bot (&lt;code&gt;flood&lt;/code&gt;). This will allow users to perform repetitive actions without flushing the recent changes feed (with only the &lt;code&gt;bot&lt;/code&gt; user right). However, I would suggest that for the pseudo-bot user group:
* It can be granted and revoked by custodians. &lt;s&gt;However, can curators add and remove pseudo-bot from their own accounts (and not others)?&lt;/s&gt;
* Users can remove themselves from it.
* A guideline might be necessary about the information and usage of it.
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:31, 14 May 2026 (UTC)

:This sounds good. Which other wiki could we model this user group on? e.g., [[b:Wikibooks:Pseudo-bots]]? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:19, 15 May 2026 (UTC)
::@[[User:Jtneill|Jtneill]] Wikiquote has a similar group: [[:wikiquote:Special:ListGroupRights]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:25, 15 May 2026 (UTC)
: Should we allow curators to add and remove themselves from the pseudobot user group (from their own account) as well? I see no objections to creating the user group. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:20, 18 May 2026 (UTC)
::My thinking is perhaps not curators by default because there should be clear visibility about their actions until they are well trusted. Let's draft a guideline or proposed policy ([[Wikiversity:Pseudo-bots]]) for the proposed user group. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:39, 18 May 2026 (UTC)
::: A solution is that they can ask any custodian to grant that group, and to remove themselves when done. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:17, 19 May 2026 (UTC)
:::: Yes, that sounds good. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:12, 19 May 2026 (UTC)
: I'll file a Phabricator task by tomorrow if there are no objections. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:01, 19 May 2026 (UTC)

== Coming over From wikinews ==
Any chance someone could help me if you are allowed to write news articles here since wikinews is going read only mode soon, thank you! [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:43, 1 May 2026 (UTC)

:The scope of Wikiversity is very broad and is basically about more-or-less any learning material. We have made it a point to not have duplicative content of other WMF projects, but since Wikinews is being shuttered, I personally am fine with writing news articles here. One thing that is not controversial at all is a learning resource &lt;em&gt;about&lt;/em&gt; how to write news: that could be hugely useful here and could involve the process of writing news stories to learn and to share back and forth with an editor or fact-checker. In fact, I'd support an entire namespace dedicated to keeping the notion of Wikinews alive here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:38, 1 May 2026 (UTC)
::Thank you so much! How do I start? Cheers! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:07, 2 May 2026 (UTC)
:::I think it's premature to start just making news articles en masse, but if you want to start discussing the topic of citizen journalism, you can do that now. [[:Category:Journalism]] already has some material, so you can start by seeing what we already have, how you can refine that, etc. You can definitely have learning resources with collaborators who want to learn about journalism ASAP. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:24, 2 May 2026 (UTC)
::::thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:38, 2 May 2026 (UTC)
::::If I could try and start one News Article could you please tell me how to go about it? Like what style of writing like Wikinews or something else? Thank you Justin! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:48, 2 May 2026 (UTC)
:::::Honestly, there are very few policies and guidelines here. I think the best way to write a news story would be in a manner that is obvious and instructive. So, for instance, it's common to use the &quot;pyramid style&quot; when you're writing news, so if you were to write a story that makes it very clear that you are using that approach, that would be helpful. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 02:08, 2 May 2026 (UTC)
::::::cool thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 02:13, 2 May 2026 (UTC)
::::::im ready to write @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 21:30, 13 May 2026 (UTC)
:::::::I think we should get more local consensus for a big project like including the entirety of the scope of Wikinews here. Again, I support it personally. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:55, 13 May 2026 (UTC)
::::::::ok lets begin. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:15, 13 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 &quot;practicing journalism&quot; 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)

===Comments and questions===
:Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice.
:Initial questions:
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
:* What are &quot;active editions&quot;?
:* 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 &quot;active editions&quot;?
::*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 &lt;em&gt;about&lt;/em&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)

== Inactivity policy for Curators ==

I was wondering if there is a specific inactivity polity for curators (semi-admins) as I am pretty sure the global policy does not apply to them as they are not ''fully'' sysops. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:20, 15 February 2026 (UTC)

:Unfortunately, I don't see an inactivity policy, but if we were to create such a new policy for curators, it should be the same for custodians (administrators). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:45, 15 February 2026 (UTC)
::@[[User:Codename Noreste|Codename Noreste]] There is currently none, that I could find, for custodians either. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:47, 17 February 2026 (UTC)
:::I think we should propose a local inactivity policy for custodians (and by extension, curators), which should be at least one year without any edits ''and'' logged actions. However, I don't know which page should it be when the inactivity removal procedure starts. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:53, 17 February 2026 (UTC)
::::@[[User:Codename Noreste|Codename Noreste]] In theory, there should be a section added at [[WV:Candidates for custodianship]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:55, 17 February 2026 (UTC)
::::: To be consistent with the [[meta:Admin activity review|global period of 2 years inactivity]] for en.wv  [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship#How are bureaucrats removed?|Bureaucrats]] we could add something like this to [[Wikiversity:Curators]]:
::::::The maximum time period of inactivity &lt;u&gt;without community review&lt;/u&gt; for curators is two years (consistent with  the [[:meta:Category:Global policies|global policy]] described at [[meta:Admin activity review|Admin activity review]] which applies for [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship|Bureaucrats]]). After that time a custodian will remove the rights.
::::: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:51, 27 March 2026 (UTC)
:::::Yup, I agree with Jtneill, there is a policy proposal for Wikiversity:Curators, where it should be logically deployed. The question is if we are ready to aprove the policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:43, 17 April 2026 (UTC)
:::::: I agree, but we should notify the colloquium about inactive curators, just like a steward would do for inactive custodians and bureaucrats per [[:m:Admin activity review|AAR]]. What is the minimum timeframe an inactive curator should receive so they can respond they would keep their rights? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:49, 17 April 2026 (UTC)
:I incorporated these suggestions into the proposed curators policy. Please review/comment/improve. Summary: 2 years, notify curator's user page, then remove rights after 1 month: [[Wikiversity:Curators#Inactivity]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:59, 24 April 2026 (UTC)
:: @[[User:Jtneill|Jtneill]] I created [[Template:Inactive curator]] for this. Feel free to make any changes or improvements. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:29, 24 April 2026 (UTC)
:::Wondering, should we also have:
:::* {{tl|Inactive custodian}}
:::* {{tl|Inactive bureaucrat}}
:::or perhaps just a single template with a parameter(s) for the user right(s)/role(s)? e.g., 
:::* if a custodian is inactive for 2 years, then custodian and curator rights are to be removed and 
:::* if a bureaucrat is inactive for 2 years, then bureaucrat, custodian, and curator rights are to to be removed.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:58, 13 May 2026 (UTC)
:::: I would probably modify that template when we actually develop our own inactivity policy, because we're currently under the AAR (a steward notifies the colloquium with [[m:Admin activity review/Notice to communities]], and inactive advanced right holders with [[m:Admin activity review/Notice to inactive right holders]]). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:16, 13 May 2026 (UTC)
:::::Ah, I see. Yes, that makes sense. Thankyou. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:21, 15 May 2026 (UTC)
: In that case, should we develop our own inactivity policy (e.g. on [[Wikiversity:Inactivity policy]] or [[Wikiversity:Support staff/Inactivity]])? I would list the general inactivity part, the process, etc. Once it's approved as a policy, I will [[m:Stewards' noticeboard|notify the stewards]]. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:30, 16 May 2026 (UTC)
::Originally, I would have thought that, for a small wiki like en.wv, it made sense to leave inactivity monitoring to the stewards. However, with the creation of the curator user group, we have already taken on local responsibility for monitoring inactivity in at least one advanced-rights group. Extending this to custodians and bureaucrats would not add much additional overhead and would provide a more consistent and transparent local administrative process.
::One option would be to develop a single, centralised policy covering all advanced-rights groups.
::An alternative would be to include an ==Inactivity== section on each relevant policy page (e.g., we already have [[Wikiversity:Curatorship#Inactivity]], but not yet in the custodianship, and bureaucratship policy pages). This approach would allow some flexibility because different user groups may warrant different criteria (such as inactivity thresholds, qualifying activity, or review procedures).
::A hybrid approach may be best: maintain separate inactivity sections within each user-group policy page, while transcluding these into a central overview page such as Codename Noreste suggests. This would preserve clarity at the local policy level while also providing a single reference point for consistency and oversight. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:09, 16 May 2026 (UTC)
::: I would suggest we develop a centralized inactivity policy page, and include a short summarized section of that page, on the support staff user group pages. We must also include a link to that policy page if we were to add &lt;nowiki&gt;== Inactivity ==&lt;/nowiki&gt; to each of those user group pages. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:48, 17 May 2026 (UTC)

== Inactive curators ==

Hello, even though [[Wikiversity:Curators]] is not a policy yet, there are curators listed here that have been inactive for two years or more:
* {{user|Cody naccarato}} (last edit on 13 Dec 2022, last logged action on 10 Dec 2022)
* {{user|Praxidicae}} (last edit on 10 Sep 2022, last logged action on 12 Sep 2022)
[[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 19 April 2026 (UTC)

:Yup, I would remove the rights. To get the rights back if theyll come back should not be a big deal. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:08, 24 April 2026 (UTC)
:: When they don't reply by May 19, feel free (or any custodian) to do so. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:28, 25 April 2026 (UTC)

==Curator inactivity review==
These curators haven't been active for &gt; 2 years. As per the [[Wikiversity:Curatorship|curatorship policy]]:
* [[Special:Log/Cody naccarato]] was notified on their talk page by [[User:Codename Noreste|Codename Noreste]] on 24 Apr 2026
* [[Special:Log/Praxidicae]] was notified on their talk page by [[User:Codename Noreste|Codename Noreste]] on 24 Apr 2026
* [[Special:Log/Tegel]] was notified on their talk page by [[User:Jtneill|Jtneill]] notified their talk page on 16 May 2026

The policy allows a month to hear from these users. If no response, a custodian will remove their curator rights.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:14, 16 May 2026 (UTC)
: For Cody naccarato and Praxidicae, their rights are to be removed by the 19th of May if they don't respond either here or on their talk page. For Tegel, the removal will happen on the 16th of June, probably. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:13, 16 May 2026 (UTC)
::Should be 24 May for Cody naccarato and Praxidicae? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:11, 16 May 2026 (UTC)
::: I made [[#Inactive curators]] on the 19th of April. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:18, 17 May 2026 (UTC)
::::OK, I see (had missed that thread, sorry - I've now moved the the 3 inactivity topics to be adjacent).
::::I'm thinking the curator policy indicates one month from user talk page notification? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:44, 17 May 2026 (UTC)
::::: Yes. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:49, 17 May 2026 (UTC)
: @[[User:Juandev|Juandev]] and @[[User:Jtneill|Jtneill]]: feel free to remove Cody naccarato and Praxidicae's curator permissions. They have not responded at all after one month. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:29, 19 May 2026 (UTC)

== [[Wikiversity:Deletion policy]] proposed as policy ==

[[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]].

This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]].

Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy.

=== Voting ===
*{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:33, 18 May 2026 (UTC)
*{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC)

=== Comments ===</text>
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== [[Korean/Words]] ==
(I go to RfD instead of ''proposed deletion'' since many pages are affected.)

I proposed to quasi-delete, i.e. '''move to userspace''' of the main (or sole?) creator, {{User|KYPark}}.

The page is organized a little bit like a dictionary. It makes it redundant to Wiktionary except that Wikiversity allows original research and there does seem to be original research there. Thus, its being organized as a dictionary would alone not necessarily be a problem.

Where I see a problem is in the organization and execution/implementation. Consider [[Korean/Words/가다]], which seems rather typical of the subpages (some subpages are like categories and transclude the pages for individual words):
* On the putative definition line, there is this: &quot;한곳에서 다른 곳으로 장소를 이동하다&quot;, apparently(?) in Korean. That does not seem to fit well into the ''English'' Wikiversity.
* There seems to be some original research into etymological relations between Korean and European languages in the &quot;Comparatives&quot; section (from what I recall, the English Wiktionary rejected this kind of content from KYPark). Admittedly, it is marked using &quot;This is a primary, secondary and/or original Eurasiatic research project at Wikiversity&quot;, so it could be tolerable, but even so, one has to wonder whether Wikiversity wants this kind of fringe science/research or outright pseudo-science.
** Fringe science: fringe physics has been moved to user space before. This would be fringe etymology. But then, original research is allowed.

Deletion is not required; moving to user space suffices, I think. Alternatively, one could at least rename the pages to make it clear from the title that this is not Wikiversity voice but rather KYPark voice, e.g. &quot;Korean/Words (KYPark)/...&quot; or  &quot;Korean/Words/KYPark/...&quot;  (recall the &quot;Fedosin&quot; pages featuring the name &quot;Fedosin&quot;).

Methodology: I see almost no methodological notes spanning the words at [[Korean/Words]]. And yet, if this is original research inventing new etymological connections, surely there should be some general considerations/analysis on how to proceed and how that manner of procedure differs from mainstream etymology?

Prefix index (max 200 items?):
{{collapse top}}
{{Small START}}
{{Special:Prefixindex/Korean/Words}}
{{Small END}}
{{Collapse bottom}}
--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:33, 24 September 2025 (UTC)

:I would keep it. If there is a course of Korean, why not to have a resesearch on Korean vocabulary? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:53, 16 October 2025 (UTC)
:: I propose to dismiss the above input: 1) it does not contain any argument, except for a question, and a question is not an argument (it can be so reinterpreted, but that includes additional burden on the interpreters, in interpreting it the wrong way); 2) it ignores all the issues I have raised, including that there is something like definition lines in Korean, in this ''English'' Wikiversity. To answer the question asked: there can be a research on Korean vocabulary in the mainspace, but not one showing the defects I identified above. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:35, 15 November 2025 (UTC)
:I've reviewed a sample of approximately 20 of the Korean/Words sub-pages and lean towards moving to user space because:
:* The pages appear to be an idiosynchratic collection of etymological pages about Korean language
:* There is minimal English instruction which is problematic for English Wikiversity
:* There is no explanation of research method
:* There is no educational rationale
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:31, 22 November 2025 (UTC)
:Well, since the original creator has indef I change my mind and I would '''delete''' it. The case is nobody knows how to continue with the research and if we move it to the userspace, the user cannot improve it eihter. What the original user can do to request admin, to send them a contentent to their email for example if they really want to improve the resource elsewhere. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:38, 11 March 2026 (UTC)

I think the consensus here is delete. {{U|Codename Noreste}} do you know an efficient way to mass delete these pages? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 17 May 2026 (UTC)
: I would use a script, but I would probably not delete those pages yet until we have the pseudobot user group. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:53, 17 May 2026 (UTC)

== [[Enhancing Web Browser Security through Cookie Encryption]] ==
{{archive top|'''Kept'''. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 17 May 2026 (UTC)}}
To avoid further conflict with the user who entered this text into Wikiversity, I am opening a RFD request.

I am not sure about how to proceed, although I am inclined to move it out of mainspace = quasi-delete. I am looking forward to get input from others, especially curators and custodians. Some considerations:

1) There is perhaps no more appearance/suspicion of copyright violation, now that the ResearchGate (RG) article (of which this is a copy, perhaps an incomplete copy?) carries a license.

2) The article is not a complete replica from RG: at a minimum, it lacks images. The inserter could have continued editing the page in his user space before he uploads images, that is, before he finalizes the page for consumption, but that did not happen. I did not check whether the text is an exact one-to-one match; the article does not indicate anything in that regard.

3) The principle implied seems to be this: users should feel free to duplicate non-peer-reviewed articles from RG in English Wikiversity, perhaps to increase the Google search and LLM yield. I find this problematic, in part for the duplication. I would say: choose a venue and publish it there. If RG is not good enough for you as a publishing venue, choose Wikiversity instead, but not both?

4) There are some features that appear unduly promotional. There is a link to a dot com home page of the inserter of the article. I dot not know how we handle or should handle this, whether prohibit such a link, etc. This is perhaps not so much a call to quasi-deletion but a call to make it less promotional.

5) I cannot determine the value of such an article. It seems to be a pseudo-article describing someone's browser extension. Can someone do a better analysis?

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:48, 8 October 2025 (UTC)

:2) Images for Wikicommons are being created, it will take a lot of time. and the text is not an exact one-to-one match
:3) I also mentioned that It was being created so that it is more accessible from mobile phone, which is not possible in RG or in Zenodo
:Let me clarify the purpose of uploading it to different platforms
:Zenodo - registration and to link DOI
:RG - Self Archiving 
:Wikiversity - Accessible by anyone from any device. LLMs may get trained on Wikiversity data or use these data for indexing 
:5) The paper is a result of a research project which involved a browser extension which was built to test the theory.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 01:34, 9 October 2025 (UTC)
:: I find the practice here of publishing non-identical but similar text (&quot;the text is not an exact one-to-one match&quot;) with almost the same title to be problematic. I cannot imagine this is a recommended practice in academic publishing. At a minimum, somewhere near the top, the page should say something like the following: &quot;This text is based on article ___ published at ___ but is not identical. The author of the differences/changes is ___.&quot; Everything else leads to an undesirable confusion. In academic publishing, the title of an article serves as key part of identification of the artifact.
:: As I said before, I seen nothing particularly academic article-like about the page except for external/superficial signs. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:30, 9 October 2025 (UTC)
:::That Article has been published under CC BY SA 4.0
:::And I am one of the author of the article. That gives me right to modify text and publish it under a similar name. However, I will add the disclaimer text that you have suggested. I hope that helps.  [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 06:07, 9 October 2025 (UTC)
:::: It may give you that right from the ''copyright'' perspective, but perhaps not from ''academic publishing integrity'' perspective. Unfortunately, I do not have any guideline handy; I am merely following my common (or not so common) sense. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:32, 9 October 2025 (UTC)
:: I would like to ask: was this article guided by someone from an academic institution, such as a university? Is it reviewed at least in some weak sense? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:39, 9 October 2025 (UTC)
:::Yes, This article has been reviewed by two academic professors, their names are also listed as co authors.
:::First, a project guide would help us with selecting a topic and with the document 
:::Second, an Internal examiner would go through our experiment and approve it
:::Finally, External Examiner would examine the documentation and verify it. 
:::We were required by these professors to put their name under contributions [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 05:48, 9 October 2025 (UTC)
:: Let me explicate the promotional potential of such a page a bit: one can go to the page of the article in Wikiversity --&gt; https://tomjoejames.com/ --&gt; HitMyTarget (a commercial, profit-making entity?) Why would the link be to a commercial web site rather than an academic page, or perhaps a LinkedIn account, which I think the person has? There could also be no link at all; a search for the name would turn out something in Google as well. But providing a direct link would drive users/viewers toward that website much stronger since otherwise the viewer of the page would have to open a new Google search window or the like. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:45, 9 October 2025 (UTC)
:::It is evident that the website is not even close to being complete. 
:::I will be creating a separate page under the same domain name specifically for people to contact me.
:::The url would probably be defined as tomjoejames.com/contact-me/
:::I haven't decided yet. But that is my personal website.
:::If the community requires me to remove it, I will. But personally I think people who are from here most likely to click the link to know more about me or to contact me. Either way I think my personal website serves the purpose. 
:::As for the HitMyTarget, it can be traced from any of my links. From my research gate profile, linkedin page or even my own userpage.
:::On the article I did not add any promotional content about myself, I hyperlinked only my own name. I do not know how that is promotional. [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 06:04, 9 October 2025 (UTC)
:::: I am pausing any further responses from me to see whether anyone else has any input. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:30, 9 October 2025 (UTC)
:What does it mean &quot;There is perhaps no more appearance/suspicion of copyright violation&quot;? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:57, 16 October 2025 (UTC)
:I have accepted VRT permission per [[ticket:2025100410001149]] FYI. [[User:Matrix|Matrix]] ([[User talk:Matrix|discuss]] • [[Special:Contributions/Matrix|contribs]]) 11:00, 28 October 2025 (UTC)
::Thank you Matrix [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 12:43, 28 October 2025 (UTC)
:I would '''delete''' it. 1) it states its a learning resource. It could not be a learning resource as not rewieved original research. 2) It is not an ongoing research, nor the research was performed on Wikiversity - wv is not a preprint or article database. Maybe it could be moved elsewhere withn Wikimedia domain, but I dont know where. So I would delete it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:56, 20 November 2025 (UTC)
::I would '''keep it.''' Like Dan had pointed out,  we do have article-like pages in Wikiversity, and this is not just a random pseudo science article but an article that is a report of an final year project, it has been reviewed by 3 professors whose name has been mentioned at the very beginning.   [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 14:50, 21 November 2025 (UTC)
:::I think it is not good to rate pages by appearance. It can be done on other Wikimedia projects, but it cannot be done on Wikiversity, because Wikiversity does not create a static format for presenting information, but is focused on the goal and process. Unfortunately, the goal and process do not have a uniform format. While a target article on Wikipedia or an entry on Wiktionary have some standard target format, Wikiversity does not. That is why I personally rate pages according to the goals and their assessment. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:05, 22 November 2025 (UTC)

Further reading for this nomination: [[S: Wikisource:Proposed_deletions/Archives/2025#Index:Cookie_Encryption.pdf]]; EncycloPetey handled the matter. Let me quote his wisdom on Zenodo (which I lack): &quot;This is tied to a PDF on Commons that was uploaded as &quot;own work&quot; with a CC license and a doi link to Zenodo, with no indication of where this paper was published or if it was published. Zenodo is not a publisher; it is a site for storing research and sharing papers. If Zenodo is the only place this was &quot;published&quot; then it was effectively self-published. --EncycloPetey (talk) 16:14, 15 September 2025 (UTC)&quot;

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:55, 9 October 2025 (UTC)

:Can you clarify what point are you trying to state? Didn't I already state that the article is published by me? 
:I first created the article in wikisource which I thought would be the perfect place, unfortunately they do not allow self published articles that are not notable. Then I discovered Wikiversity where they allow self published articles. That is why I created the article here. 
:Unlike in wikisource, I did follow guidelines.
:Ever since you deleted the first article, I spent time reading Wikiversity guidelines and I do think that I am following it perfectly.
:I would like to get your suggestions on how should I improve the page,  10 points would be sufficient. 
:Because your goals or intentions are confusing me very much. At first you told me that the article is exactly the same as the preprint in RG and therefore there is no use to it here. And then when I continued to optimize it for Wikiversity, you went ahead and said it is problematic according to recommended academic publishing. 
:Atleast just respond to the points that I have made whether you agree or disagree. So that I clarify and proceed to discuss points that are important and relevant 
:Have you published an research article? If yes, could you send it to me so that I can see the format you have done it [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 10:45, 9 October 2025 (UTC)
:: I am giving a chance/time to other curators/custodians to look at the matter and respond to my inputs. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:14, 9 October 2025 (UTC)
:: Incidentally, above I counted 4 questions (or more), 1 request (or more?) and 1 command (or more?). That is a behavior of a commanding entity. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:24, 9 October 2025 (UTC)

I would '''delete it''''. It's more like an academic communication than a learning resource or research.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:32, 26 October 2025 (UTC)
:: In the above post, I do not see any valid rationale for deletion: we do have article-like pages, in Wikijournals and also e.g. in [[Physics/Essays/Fedosin/Stellar Stefan–Boltzmann constant]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:59, 3 November 2025 (UTC)
:::But I do, see above. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:56, 20 November 2025 (UTC)

:it is a '''student research paper''' forming part of a learning resource on web security and encryption.
:The project was conducted as part of a final-year university course and documented as a practical study on cookie encryption and it has been reviewed by three professors. However, I will be creating a sub page for the article to elaborately describe the experiment that we have conducted and the results we got.    [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 15:57, 26 October 2025 (UTC)
::And why should w host research papers? Wikiversity is not an academic Journal nor repository. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:06, 22 November 2025 (UTC)
:::I do not wish to go through this same argument once again, I've already answered to this question several times in Dan's talk page, Colloquium. you can refer them. I am not hosting the research paper here, I have already hosted the pdf in the ResearchGate, I have published a text version in the wikiversity so that it may be useful for others. Unless you can show me how that article is totally useless, I would like to '''keep''' the article in the wikiversity.    [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 10:13, 22 November 2025 (UTC)
::::And thats the point I am having. Wikiversity is not paper repository. The only way is to publish it via WikiJournal, but they want it for Wikipedia usually. Why wikiversity should be a duplication of ResearchGate, Academia or Zenodo?
::::What I can read on [[Wikiversity:What is Wikiversity?]] policy is, that Wikiversity research &quot;...includes interpreting primary sources, forming ideas, or taking observations.&quot; The article doent look to fall into this. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:43, 22 November 2025 (UTC)
:::::Well, then how come you missed the term &quot;Learning Projects&quot;? As Jtneill had pointed out, this is a legitimate learning project. And also, I do have the VRT permission to host this article on Wikiversity.  [[ticket:2025100410001149]] . besides ResearchGate is an self-archiving platform. the document version in it is not accessibly to screen readers (usually disable people use them), Translators, and also for the mobile readers. therefore I do have valid reasons to publish this article on wikiversity. 
:::::# It is a learning project, therefore according to WIkiversity Policy, It qualifies. 
:::::# I have an explicit VRT permission to host this article on Wikiversity
:::::# Versions that are published in RG, Zenodo are documents, and they are not accessible by screen readers or mobile users. Therefore it is imperative that an article version of this paper exist on here. 
:::::Therefore this article qualifies to stay here on Wikiversity.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 11:22, 22 November 2025 (UTC)

'''Keep'''. This is a legitimate student learning project that may be of use to others. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:51, 22 November 2025 (UTC)
{{archive bottom}}

== [[Pragmatics/History]] ==
{{archive top|Deleted. Other related resources have been deleted. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:24, 17 May 2026 (UTC)}}
Another KYPark page and subpages with unclear organization scheme. Contains fairly many redlinked items. See also [[User:KYPark/Literature]], perhaps a similar concept. Unlikely to be really useful for others but KYPark. '''Move to user space'''.

As an alternative, moving to [[History of Pragmatics (KYPark)]] would make sense to me: the topic is identified using a natural-language phrase (instead of the relatively unnatural slash) and the responsible editor is indicated so that the reader knows whether to look or not. And for those who oppose the brackets (which I like): [[History of Pragmatics/KYPark]]. Or also: [[KYPark/History of Pragmatics]]. But then, searches in mainspace will see that content and the question is whether that is good. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:21, 15 October 2025 (UTC)

:What about to propose the user to write some guidelines, how other can participate instead of deleting it? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:03, 16 October 2025 (UTC)
:: I plan to move the pages to userspace as I proposed. If someone wants to ask KYPark to address the problems, they should feel free. There will be plenty of time for KYPark to address the problems while the material is in user space. After the problems are addressed, the material can be moved back to mainspace. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:38, 15 November 2025 (UTC)
:So I would '''delete''' it. In the blocked user space its useless. The user cannot improve it and Wikiversity is not free hosting service for personal pages. My believe is, that there should be just a few working pages in the users spaces. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:30, 11 March 2026 (UTC)

'''Move'''. Insufficient statement of learning objective or connection to related learning resources with insufficient current activity to stay in main space. The page was originally [[History of pragmatics]] but was moved by Dave B. Therefore, I suggest moving to [[User:KYPark/History of pragmatics]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:57, 22 November 2025 (UTC)
{{archive bottom}}

== [[IMHA Research Archives]] ==
I propose to '''move to userspace''', including the subpages. I struggle to understand how Wikiversity readers are supposed to benefit from the material here and in the subpages. In the log, there is e.g. '10 February 2019 Marshallsumter discuss contribs deleted page IMHA Research Archives (content was: &quot;{&lt;nowiki/&gt;{Delete|Author request}} Thanks! -&quot;)', so the page was deleted before, but not the subpages.

We could also delete all the material if we have strong enough suspicion too much of it is copyright violation. In any case, moving to user space improves the matter a little by moving the content away from Google search. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:38, 9 November 2025 (UTC)

:Looking at some sub-pages, they can be deleted e.g., because they only consist of broken links or are largely empty. I deleted a couple but haven't been through all to check. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:27, 10 November 2025 (UTC)

As an example, let me give the wikitext content of [[IMHA Research Archives/3. Scientific litterature search, storage and use]]:
&lt;pre&gt;
==[[/Medicina Maritima - the Spanish scientific maritime health journal/]]==

==[[/PubMed/]]==

==[[/Google and Google Scholar/]]==

==[[/Zotero/]]==

==[https://www.dropbox.com/sh/d91z7bcyelfvk42/AAAkIvjtBnnFMbiU9ZLOdVL9a/Andrioti_database%20sources0310.pptx?dl=0 Maritime health web portal ressources ]==
&lt;/pre&gt;
The wikilinks are red; the external link to dropbox says &quot;You don't have access&quot;. This was made in 2016. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:04, 11 November 2025 (UTC)

:I suggest delete -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:27, 12 November 2025 (UTC)
:: I think we should avoid deletion as much as possible, instead moving to user space (bar copyvio, ethics violation, etc.). This is a good general principle. It greatly improves auditability and makes it so much easier for anyone to request undeletion since they know what content they are requesting for undeletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:52, 12 November 2025 (UTC)
:::Do not recreate Wikiversity from the educational and research project to the personal blog. That will lead to the cancelation of it by WMF. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:44, 20 November 2025 (UTC)
:::: The English Wikiversity has a long tradition of moving problematic content to user space, as per evidence collected at [[User:Dan_Polansky/About Wikiversity#Moving pages to userspace]]. If Wikimedia Foundation finds this problematic, they can start a discussion in Colloquium and state their concerns. They do not need to make explicit threats at first; they can start a discussion and explain why it is problematic. They can even do it from an anonymous IP and provide a well-articulated reasoning. And anyone else can start a discussion in Colloquium to change this tradition. I do not see why we should not want to change that tradition based on well-articulated, compelling reasoning. I see no reason why Juandev should be making threats instead of them, on a per RFD basis. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:58, 21 November 2025 (UTC)
:::: If Juandev is ''sincere'' about deleting very-low-value items ''from user space'', he should perhaps demonstrate that by asking his pages like [[:cs:Uživatel:Juandev/Problémy/Kov/Repase dvířek elektroskříně]] to be deleted; otherwise, I register a ''glaring inconsistence''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:43, 21 November 2025 (UTC)
::What was the original delate page about @[[User:Jtneill|Jtneill]]? I guess that would be crucial for the decission. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:48, 20 November 2025 (UTC)
:::@[[User:Juandev|Juandev]] the couple of pages I checked and deleted were much like @[[User:Dan Polansky|Dan Polansky]] posted above i.e., headings with empty sections and/or broken links but no substantive content. But I think each sub-page needs checking. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:59, 20 November 2025 (UTC)
::::So I'm saying that the main page usually determines what the other pages are for. But if I don't know the page because it's been deleted, or why was deleted (deletion based on the founder's request is probably not the rule), it's hard to judge. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 22:16, 20 November 2025 (UTC)
:::::I've pasted the original content of the root page: [[IMHA Research Archives#Original page]] (i.e., prior to the content being removed and deletion requested) to help understand the context for the sub-pages. In 2018, Saltrabook blanked the page, indicating that the content had been moved elsewhere, and requested page deletion. Marshallsumter then deleted the main page but not the sub-pages. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:58, 21 November 2025 (UTC)
::::::I see, so if those subpages are usefull I would keept them, if not I would delete them. I dont see a point of providing free hosting to sombody, by moving many pages to their user space. The question is if we want to host (i.e. to have in the main ns) lists of links elsewhere. I have no opinion on that. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:11, 22 November 2025 (UTC)

: Let me clarify that while many of the subpages are like the example above, [[IMHA Research Archives/Scientific litterature search, storage and use/Zotero]] is different:
:: &quot;A continuous critical and evidence based learning is a core issue in clinical practice, research, teaching, publication and prevention activities. The Zotero Program is just one of many scientific literature management programs, that should be used for these purposes. Of course one can live without such a database but it helps a lot and can save a lot of time that could be used for more interesting issues. Not only that, but it helps to create better publications and knowledge. Without this program it can be very time consuming to publish a scientific article with the requested style for the references. Further in daily practice when you want to collect and cite a few references for a specific evidence in a clinical colloquium and discussion, this program is excellent. Therefore we strongly recommend that all maritime health persons learn how to use this excellent tool in their daily maritime health practice of all different types. There are good online courses for self-instruction on how to use Zotero. For example this one: Zotero fast online course But in order to increase IMHAR´s collective scientific strength in the use of EBM we would like to give training sessions in every possible opportunity, IMHA Symposia, seminars and other types of meetings. The database is useful for personal purposes but especially also for collaborative aims. At the IMHAR meeting in Paris Oct 7th 2016 we will give an introduction to the program by showing how it can be used in the daily practice and discuss strength and weaknesses compared to other similar databases.&quot;
: Even longer is e.g. [[IMHA Research Archives/Scientific litterature search, storage and use/Medicina Maritima - the Spanish scientific maritime health journal]].
: However, that does not mean these should be salvaged. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:53, 21 November 2025 (UTC)
:{{ping|Saltrabook}} I'm wondering if you can respond here to help us decide about whether to delete the IMHA Research Archives sub-pages or perhaps move them to your user space? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:58, 17 May 2026 (UTC)

== [[Palliative medicine]] ==
{{archive top|'''Kept'''. Page has been improved. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:38, 17 May 2026 (UTC)}}

Underdeveloped and has not been improved on since 2007. Author inactive. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:42, 14 December 2025 (UTC)

:Delete, per nominator [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:16, 22 January 2026 (UTC)
:Yes, I would also expect there to be more and especially that someone would write how to use it. However, it still seems to me to be a useful thing in the sense of a syllabus, so that someone who is interested in the topic knows what information to obtain in order to get a complete picture of the topic. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:55, 16 March 2026 (UTC)
{{archive bottom}}

== [[Canadian Wilderness]] ==
{{archive top|Deleted per nom. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:07, 17 May 2026 (UTC)}}
This page doesn't seem to belong to wikiversity. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 09:55, 6 February 2026 (UTC)

:In principle there could be some material useful here but in practice, I don't see what this page is adding as an educational resource. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 12:54, 6 February 2026 (UTC)
:I can see this being a useful resource to a bigger project. Maybe we could move it to the &quot;[[Wikiversity:Drafts|Draft]]&quot; namespace vs. deleting it? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:28, 6 February 2026 (UTC)
::Does anyone plan to work on it? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 01:59, 8 February 2026 (UTC)
:::Next week the page has it's 17th birthday. Ever now and than someone added to it. With a lot of work it could be a nice encyclopedic article but making it educational .... Merging it may take more work than rewriting it. Move to Draft might be the best option. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 08:58, 12 February 2026 (UTC)
{{archive bottom}}

== [[LQR Control for an Inverted Pendulum]] ==
{{archive top|'''Deleted''' per nom. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:36, 17 May 2026 (UTC)}}
Underdeveloped resource, has not been edited for more than a decade. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:03, 16 March 2026 (UTC)

:Looks like a test, '''delete'''. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:30, 27 March 2026 (UTC)
{{archive bottom}}

== False flag &quot;authority hack&quot; user page deletion ==
{{archive top|'''Not undeleted''', the requester dropped the request. See Wikiversity:Requests for Deletion v. 2803217.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:36, 10 May 2026 (UTC)}}
'''Undeletion requested'''

Hi, Juandev marked my user page as &quot;spam&quot; and &quot;authority hack&quot;, and deleted it. 

First, I asked him for help with &quot;time limit for new users&quot;, and he replied - I should admit I dont know, what is &quot;new user limit&quot;, but if filter blocks your page because of certain external link, you may force to save anyway and sometimes it works. It should not work, when the website is blacklisted. As of now, I am not seeing you to save page in main namespace, so try to save it without external links first. 

Then he wrote me another message: Well, I have analyzed your contribution to Wikiversity and I should point out here, that this project is not a place for advertising, so there is no way of promoting your books and authority this way. - probably referring to the intro of my About me page where I present me and my work. 

Before I could explain him the difference between the neutral information and advertising and promotion, he deleted my user page. 

Here is my answer I posted to the discussion today: 

: Hi, my About Me page is just an info page with the neutral as possible presentation of my work.
:
: There is a big difference between informing and advertising. Informing is neutrally stating that something exists and requiring no action, while advertising is a special communication form with intent to cause certain action from readers. For example, click here, click there, order this, buy that.
:
: There is no such intention, form, or terms on my info page. Just neutral information. I don't hide and I am not ashamed that I am write and author, and that is a part of the usual bio, including works. I checked your user page: &quot;I graduated from the Czech University of Life Sciences in Prague and studied information science at the Faculty of Arts of Charles University.&quot; I think that if you had written a book on Life Science, you would have mentioned that as well.
:
: Most of the Info page is about my research and AIPA Method which is a valid contribution to psychology, consciousness studies, identity theory, and personality development. Actually, my paper '''AIPA Method: A Cognitive-Phenomenological Model for Identity Reconstruction and Stabilization in Pure Awareness''' is now in the peer review procedure at Journal of Consciousness Studies.
:
: Here is a part from the Wikiversity AIPA Method page in creation (waiting for the end of the time limit for new users):

: == Introduction ==
: The AIPA Method addresses a gap in contemporary personal development and consciousness science: most evidence‑based approaches (CBT, MBSR, MBCT, standard meditation) operate at the level of mental content—reframing thoughts, observing them, or reducing their impact—rather than at the level of identity structure. In contrast, AIPA targets the structural relationship between the self and the mind, aiming at durable identity reconstruction rooted in Pure Awareness rather than symptom management.
:
: The central research question of the primary AIPA preprint is whether a structured, sequentially staged method can produce permanent identity reconstruction rooted in Pure Awareness, and how such a method compares to established approaches in scope, mechanism, and outcome.
:
: == Theoretical foundations ==
: The AIPA framework is grounded in the cognitive‑phenomenological tradition (e.g., McAdams, Varela, Metzinger, Erikson), contemporary consciousness science on minimal phenomenal experience, and qualitative methods advocacy in psychology. It builds directly on:
:* Empirical work on pure awareness and Minimal Phenomenal Experience (MPE), especially Gamma &amp; Metzinger’s large‑scale study of content‑reduced awareness states.
:* Metzinger’s proposal of minimal phenomenal experience as an entry point for a minimal unifying model of consciousness.
:* Narrative identity and partial‑self models within personality and identity theory.
: Within this backdrop, AIPA proposes Pure Awareness as a distinct, operationally specified state that can become a structural ground of identity rather than a transient meditative experience.
:
: == Experiential empiricism ==
: The empirical foundation of the AIPA Method is explicitly first‑person and experiential, combining:
:* A 22‑year longitudinal autoethnographic self‑study (2003–2025) documenting partial personality episodes, protocol use, and outcomes.
:* A 13‑year prospective verification period with zero self‑reported recurrence of targeted harmful behaviors after a dated stabilization point (1 January 2006).
:* A high‑ecological‑validity “stress test” during acute bereavement, used to examine whether non‑reactive awareness remains stable under maximal provocation.
:* Two independent practitioner cases (an Amazon‑verified report and a structured questionnaire case) providing preliminary convergent signals across cognitive, emotional, behavioral, and identity dimensions.
:
: All central constructs (Pure Awareness, partial personalities, the Switch, identity stabilization) are operationalized with explicit phenomenological and behavioral criteria intended to enable replication and future third‑person measurement.
:
: I believe this is a valid contribution to Wikiversity.
:
: Best regards, Senad [[User:Senad Dizdarević|Senad Dizdarević]]


I suggest you check the deleted user page, and see for yourself if it is &quot;spam&quot; and &quot;authority hack&quot;, or a legit author's page with one paragraph short presentation, while the rest of the page is about my research project. 

Thank you for undeleting my user page, so I can use it. 

Best regards, 

Senad Dizdarević [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:26, 2 April 2026 (UTC)

:Hi Senad,
:Welcome to Wikiversity.
:It looks like you tried adding similar content to Wikipedia and ran into similar difficulties over there ([[w:User talk:Senad Dizdarević]])? Perhaps that is what has led to you Wikiversity?
:Basically, if you'd like to collaborate and help build open educational resources, feel free to contribute to Wikiversity. But if the primary motivation is to promote your autobiographical work you're probably going to run into challenges.
:Sincerely,
:James
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:11, 3 April 2026 (UTC)
::James, Hi, and thank you for your answer. 
::Yes, in 2025, I created the autobiographic page on Wikipedia, which was removed because of the links to my books on Amazon. To admin, I explained that I did not know the rules, and agreed that page is removed. Now I know that somebody else must write a Wikipedia page for you. 
::On the deleted user page on Wikiversity, there were no links to Amazon or any other form of promotion, just neutral as possible basic presentation of my writing (one sentence) and current project (the rest of the page). 
::I created Wikiversity page to present my AIPA Method project, to invite researchers to read it, give their opinion, and conduct empirical researches in their institutions. Now, it is in a theoretical phase, and needs more empirical testing. 
::Best regards, 
::Senad [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:03, 3 April 2026 (UTC)
:::It looks to me like the primary motivation for contributing to Wikiversity is to drive traffic / search engine ranking to your website?
:::* [[User:Senad Dizdarević]]
:::* [[AIPA Method]]
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:36, 4 April 2026 (UTC)
::::No, it is not. There is no link to my website, so &quot;driving traffic to my website&quot; is not possible. 
::::For your educational purposes: 
::::Copilot &quot;AI:  [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:38, 4 April 2026 (UTC)
:::::So do you still insist of undeleting your former version of your userpage if you have created the new one? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:15, 6 April 2026 (UTC)
::::::No, because in the moment of undeletition, somebody could delete it again, and so on. Thank you for not deleting my new user page, as it is made in your user page image.  [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 08:59, 6 April 2026 (UTC)

{{archive bottom}}

== Undeletion request ==

It was deleted by an admin without discussion and with untrue rationale. If people take offense with the question that doesn't mean it's not a valid question and the page was good. Please undelete the Wikidebate page [https://web.archive.org/web/20250810030352/https://en.wikiversity.org/wiki/Is_it_likely_that_Earth_has_been_visited_by_aliens_millions_of_years_ago%3F Is it likely that Earth has been visited by aliens millions of years ago?]

There are lots of sources on the subject, the wikidebate is sourced very well compared to other wikidebates and wikiversity pages, and the page is educational, useful and of good quality. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 23:57, 10 April 2026 (UTC)

:Page: [[Is it likely that Earth has been visited by aliens millions of years ago?]]
:Ping: [[User:Atcovi|Atcovi]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:21, 11 April 2026 (UTC)
:There is no need for a discussion for straight garbage-level, pseudoscientific content.
:For '''Is it likely that Earth has been visited by aliens millions of years ago?''', the flaws for this page wouldn't even take someone more than a few minutes to assess:
:* Essentially, the &quot;pro&quot; arguments unproven claims being derived from irrelevant, established facts (basically: &quot;it is likely aliens have came because Earth has existed for so long [sources proving Earth's longevity]&quot;). These are not serious, scientifically-backed arguments - these are non sequiturs. It's as if I said Wikipedia has existed longer than my existence on Earth ([https://d1wqtxts1xzle7.cloudfront.net/74351725/eyJoIjogImZiODhmYzNkODU1N2UxMWExYzUyODJiYzgzZTRmZDM4OTBjODY5YWMzMjA3NDNmOWEyZTA0ZTU3ZGYwZjAyYTkiLCAidSI6ICJodHRwczovL3B1cmUuaHZhLm5sL3dz-libre.pdf?1636354596=&amp;response-content-disposition=inline%3B+filename%3DCritical_Point_of_View_A_Wikipedia_Reade.pdf&amp;Expires=1775872055&amp;Signature=GqbUZboYRvUYWi~aW40LT5eZSHrLuDL3o0-DxAH8vSvcJcGAuyByZWLF2oHTY6GlB72TqvZxpE-v9d4gvsA6myriYqO~QQQZgWxjT2JXjUWC-yiPcTF4l~lroJSi4dY0v9eKiBcU03l-aeUdrX8~UPfi0TfW0IhsmzH-VBR6X6FrzRpIqc6uM6n9YXfr5FRB3aCqqokU690af3n0Hguaub1Zgmh9qjYYqzBS0VOOHjKTTEQnDuadX3jl5CQeXYTaeCC3H0hMeVwHlratbrnuFEKC1aN0-5znCUoSzMEg21ECzGPTrSDM1W05dcK-u0ZTCeUGKAuC-2yRFL3sY46MIw__&amp;Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA#page=157 reputable source proving ''this'' fact]), therefore it's likely that my birth took place solely for the sake of me experiencing Wikipedia (0 backing). It makes no sense and no person with at least a high school-level of intelligence would take this seriously. 
:* What is worse is that the user is being misleading with their &quot;[the page is] sourced very well&quot; claim. The sources ''themselves'' don't even back up the claims. It's just used as proof for an established concept, where the user then uses this established concept to jump to an unsupported, laughable conclusion that is pulled out of thin air. It's utterly ridiculous to even consider such a page for mainspace since it clearly violates our [[Wikiversity:Verifiability]] policy. This is, once again, pseudoscientific content that has caused our website to reduce in quality over the last few years. 
:* Going source by source, we can see that:
:#[https://web.archive.org/web/20250918011642/https://timesofindia.indiatimes.com/blogs/thebigd/compress-earths-history-into-24-hrs-humans-came-at-1158-pm-yet-killed-70-of-wildlife/ ‘Compress Earth’s history into 24 hrs. Humans came at 11:58 pm, yet killed 70% of wildlife’] is literally just a blog post which doesn't even mention aliens or extraterrestrial life. It just talks about Earth's history in accordance with the 24-hour metric of time, and the author tries to use this article as a 'piece in the puzzle' of aliens &quot;possibly&quot; visiting Earth... which, once again, is unsupported and is not backed up anywhere in the article.
:#[https://web.archive.org/web/20250808053249/https://news.cornell.edu/stories/2023/11/jurassic-worlds-might-be-easier-spot-modern-earth The Cornell article does not even remotely support the idea that &quot;aliens visited Earth&quot;]. It mentions a ''chance'' of &quot;life there [a habitable exoplanet] might not be limited to microbes, but could include creatures as large and varied as the megalosauruses or microraptors that once roamed Earth.&quot;, but again, no justification to take this article as proof that &quot;aliens may have visited us!&quot;. There's no mention of aliens visiting Earth anywhere in the article. Once again this is only proving the background premise, but not the unsupported, nonsensical &quot;alien likelihood&quot; argument that the author of this garbage page is trying to push so desperately.
:#The Parker Solar Probe WP article does not even mention aliens either. It follows the same issues as the previous argument.
:And the other page this user complained about [https://en.wikiversity.org/wiki/User_talk:Atcovi#Deletion_of_educational_page_because_of_personal_opinion on my talk page] holds almost similar, maybe even more fatal mistakes, than this one. It has nothing to do with &quot;taking offense&quot;, this is just low-quality, garbage content. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:56, 11 April 2026 (UTC)
::Why do you think pro claims are required to be proven? It's possible to object to them and these are arguments, not contextualized to be statements of proven facts. And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future. Instead of insulting others' intelligence, maybe engage with the actual reasoning rather than censoring it away. And there are lots of sources, such as [https://interestingengineering.com/science/alien-civilizations-may-have-visited-earth-millions-of-years-ago-study-says Alien Civilizations May Have Visited Earth Millions of Years Ago, Study Says] etc etc. The sources are used for the arguments themselves individually. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 12:30, 11 April 2026 (UTC)
:::Because, once again, this is not a site that caters to rampant debating for the sake of &quot;we need to employ rationality and logic to solve the world's problems&quot;, we have policies that we need to fulfill. The claims made in the pro argument clearly do not meet [[Wikiversity:Verifiability]], since you cannot verify these arguments with the sources because they are not relevant. 
:::''&quot;And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future.&quot;'' The point being is that these arguments are not supported by the sources. Even the article you mention poses the idea as a hypothetical model. This is just you twisting the article to fit your unsupported narrative. I'll bring direct quotes for you to show why the linked article does not help you:
:::* ''One problem the researchers do make sure to point out is that '''they are working with only one data point: our own behaviors and capabilities for space exploration'''. “We tried to come up with a model that would involve the fewest assumptions about sociology that we could,” Carroll-Nellenback told Business Insider. '''We have no real way of knowing the motivations of an alien civilization'''.'' --&gt; proves that this is just speculation and no evidence-based arguments have been provided for the idea that aliens likely visited Earth.
:::And I'm not sure if you read my entire response, but I ''did'' engage with your &quot;actual reasoning&quot; and exposed its weaknesses and lack of adherence to Wikiversity policies. If we allowed content that was just filled with non sequiturs we would have content that fails Wikiversity's educational objectives and reduces the overall quality of this website, hence why such a harsh stance needs to be taken. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:50, 11 April 2026 (UTC)
::::Thanks for proving that the Wikimedia ecosystem is unfit to deliberate on controversial topics. The question is entirely valid and the content is far better sourced than nearly all Wikidebates and has no genuine flaws. The only possible issue with it as far as I can see is that now that Wikidebates has been paused people can't add objections if they do have sth specific to say about the topic that's not already included on that page which already had plenty of Cons and objections.
::::The page was more educational than most of Wikiversity and it was well-sourced – wikidebates was for arguments so people were invited to make arguments based on sourced things or outlined logic and the page met [[WV:V]] and most pages on Wikiversity aren't sourced as good. Doesn't look like people can see beyond their biases and personal views here but that's more evident in the marginalization and deletion of wikidebates and the low activity in that project than these selective deletions. A constructive thing to do would be to add reasoned Cons and objections not yet on the page and people had plenty of time to do that. There are and will be other sites for free constructive rational adversarial deliberation (not a big loss in that sense). [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 16:31, 22 April 2026 (UTC)
:::::Thank you for failing to address any of my arguments and going on an unrelated, nonsensical tangent that has nothing to do with the discussion. Once you start producing work that aligns with Wikiversity's content policies instead of typing up laughable, pseudoscientific garbage, then maybe your work can be accepted and not removed. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:59, 22 April 2026 (UTC)
::::::I suggest you stop ridiculing things and learn respectfully forming genuine points about the subject at hand. {{tq|the idea as a hypothetical model}} but please learn first about what arguments are and why they're not the same as a statement of objective proven fact. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 17:18, 22 April 2026 (UTC)

==Pages by Harold Foppele==
[[User:Harold Foppele]] is locally blocked indefinitely and globally banned for sockpuppetry. There were also WMF and local community concerns expressed about copyright violation and AI (over)use. As a result, I think the Wikiversity pages created by this account warrant review with regard what should be deleted, what should be retained etc.:
* [[Completing the square]]
* [[Number of independent spatial modes in a spherical volume]]
* [[Quantum]]
** [[Quantum/Andrew N. Jordan]]
* [[Quantum A Matter Of Size]]
* [[Quantum A Spooky Action at a Distance]]
* [[Quantum: A Walk Through the Universe]]
* [[Quantum Computing Algorithms in the NISQ Era]]
* [[Quantum Formulas Collection]]
* [[Quantum harmonic oscillator]]
* [[Quantum Matter Elements and Particles]]
* [[Quantum mechanics]]
** [[Quantum mechanics/Timeline]]
* [[Quantum mechanics learning module]]
* [[Quantum mechanics measurements]]
* [[Quantum Noisy Qubits]]
* [[Quantum optics beam splitter experiments]]
* [[Quantum: The Secret of Cohesion: How Waves Hold Matter Together]]
* [[Quantum Ultra fast lasers]]
* [[Speed of sound experiments]]
* [[User:Harold Foppele]]
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:12, 17 May 2026 (UTC)

:'''Delete all''' Not worth keeping. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:27, 17 May 2026 (UTC)

== [[Classical guitar pedagogy]] ==

According to the talk page, the author of this page intended to create this page for Wikipedia. At this moment in time (nearly 20 years later), the page is still riddled with red links and doesn't seem to fit Wikiversity's learning modules. Therefore, I propose that this page should be deleted. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:03, 19 May 2026 (UTC)</text>
      <sha1>1qzxnvtr9243bntoqik280fwk9czj4x</sha1>
    </revision>
    <revision>
      <id>2810422</id>
      <parentid>2810419</parentid>
      <timestamp>2026-05-19T13:05:50Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* Film writing */ new section</comment>
      <origin>2810422</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="57918" sha1="66gignailrf8meb1tad97kedqn8ew52" xml:space="preserve">{{/header}}
== [[Korean/Words]] ==
(I go to RfD instead of ''proposed deletion'' since many pages are affected.)

I proposed to quasi-delete, i.e. '''move to userspace''' of the main (or sole?) creator, {{User|KYPark}}.

The page is organized a little bit like a dictionary. It makes it redundant to Wiktionary except that Wikiversity allows original research and there does seem to be original research there. Thus, its being organized as a dictionary would alone not necessarily be a problem.

Where I see a problem is in the organization and execution/implementation. Consider [[Korean/Words/가다]], which seems rather typical of the subpages (some subpages are like categories and transclude the pages for individual words):
* On the putative definition line, there is this: &quot;한곳에서 다른 곳으로 장소를 이동하다&quot;, apparently(?) in Korean. That does not seem to fit well into the ''English'' Wikiversity.
* There seems to be some original research into etymological relations between Korean and European languages in the &quot;Comparatives&quot; section (from what I recall, the English Wiktionary rejected this kind of content from KYPark). Admittedly, it is marked using &quot;This is a primary, secondary and/or original Eurasiatic research project at Wikiversity&quot;, so it could be tolerable, but even so, one has to wonder whether Wikiversity wants this kind of fringe science/research or outright pseudo-science.
** Fringe science: fringe physics has been moved to user space before. This would be fringe etymology. But then, original research is allowed.

Deletion is not required; moving to user space suffices, I think. Alternatively, one could at least rename the pages to make it clear from the title that this is not Wikiversity voice but rather KYPark voice, e.g. &quot;Korean/Words (KYPark)/...&quot; or  &quot;Korean/Words/KYPark/...&quot;  (recall the &quot;Fedosin&quot; pages featuring the name &quot;Fedosin&quot;).

Methodology: I see almost no methodological notes spanning the words at [[Korean/Words]]. And yet, if this is original research inventing new etymological connections, surely there should be some general considerations/analysis on how to proceed and how that manner of procedure differs from mainstream etymology?

Prefix index (max 200 items?):
{{collapse top}}
{{Small START}}
{{Special:Prefixindex/Korean/Words}}
{{Small END}}
{{Collapse bottom}}
--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:33, 24 September 2025 (UTC)

:I would keep it. If there is a course of Korean, why not to have a resesearch on Korean vocabulary? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:53, 16 October 2025 (UTC)
:: I propose to dismiss the above input: 1) it does not contain any argument, except for a question, and a question is not an argument (it can be so reinterpreted, but that includes additional burden on the interpreters, in interpreting it the wrong way); 2) it ignores all the issues I have raised, including that there is something like definition lines in Korean, in this ''English'' Wikiversity. To answer the question asked: there can be a research on Korean vocabulary in the mainspace, but not one showing the defects I identified above. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:35, 15 November 2025 (UTC)
:I've reviewed a sample of approximately 20 of the Korean/Words sub-pages and lean towards moving to user space because:
:* The pages appear to be an idiosynchratic collection of etymological pages about Korean language
:* There is minimal English instruction which is problematic for English Wikiversity
:* There is no explanation of research method
:* There is no educational rationale
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:31, 22 November 2025 (UTC)
:Well, since the original creator has indef I change my mind and I would '''delete''' it. The case is nobody knows how to continue with the research and if we move it to the userspace, the user cannot improve it eihter. What the original user can do to request admin, to send them a contentent to their email for example if they really want to improve the resource elsewhere. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:38, 11 March 2026 (UTC)

I think the consensus here is delete. {{U|Codename Noreste}} do you know an efficient way to mass delete these pages? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 17 May 2026 (UTC)
: I would use a script, but I would probably not delete those pages yet until we have the pseudobot user group. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:53, 17 May 2026 (UTC)

== [[Enhancing Web Browser Security through Cookie Encryption]] ==
{{archive top|'''Kept'''. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 17 May 2026 (UTC)}}
To avoid further conflict with the user who entered this text into Wikiversity, I am opening a RFD request.

I am not sure about how to proceed, although I am inclined to move it out of mainspace = quasi-delete. I am looking forward to get input from others, especially curators and custodians. Some considerations:

1) There is perhaps no more appearance/suspicion of copyright violation, now that the ResearchGate (RG) article (of which this is a copy, perhaps an incomplete copy?) carries a license.

2) The article is not a complete replica from RG: at a minimum, it lacks images. The inserter could have continued editing the page in his user space before he uploads images, that is, before he finalizes the page for consumption, but that did not happen. I did not check whether the text is an exact one-to-one match; the article does not indicate anything in that regard.

3) The principle implied seems to be this: users should feel free to duplicate non-peer-reviewed articles from RG in English Wikiversity, perhaps to increase the Google search and LLM yield. I find this problematic, in part for the duplication. I would say: choose a venue and publish it there. If RG is not good enough for you as a publishing venue, choose Wikiversity instead, but not both?

4) There are some features that appear unduly promotional. There is a link to a dot com home page of the inserter of the article. I dot not know how we handle or should handle this, whether prohibit such a link, etc. This is perhaps not so much a call to quasi-deletion but a call to make it less promotional.

5) I cannot determine the value of such an article. It seems to be a pseudo-article describing someone's browser extension. Can someone do a better analysis?

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:48, 8 October 2025 (UTC)

:2) Images for Wikicommons are being created, it will take a lot of time. and the text is not an exact one-to-one match
:3) I also mentioned that It was being created so that it is more accessible from mobile phone, which is not possible in RG or in Zenodo
:Let me clarify the purpose of uploading it to different platforms
:Zenodo - registration and to link DOI
:RG - Self Archiving 
:Wikiversity - Accessible by anyone from any device. LLMs may get trained on Wikiversity data or use these data for indexing 
:5) The paper is a result of a research project which involved a browser extension which was built to test the theory.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 01:34, 9 October 2025 (UTC)
:: I find the practice here of publishing non-identical but similar text (&quot;the text is not an exact one-to-one match&quot;) with almost the same title to be problematic. I cannot imagine this is a recommended practice in academic publishing. At a minimum, somewhere near the top, the page should say something like the following: &quot;This text is based on article ___ published at ___ but is not identical. The author of the differences/changes is ___.&quot; Everything else leads to an undesirable confusion. In academic publishing, the title of an article serves as key part of identification of the artifact.
:: As I said before, I seen nothing particularly academic article-like about the page except for external/superficial signs. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:30, 9 October 2025 (UTC)
:::That Article has been published under CC BY SA 4.0
:::And I am one of the author of the article. That gives me right to modify text and publish it under a similar name. However, I will add the disclaimer text that you have suggested. I hope that helps.  [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 06:07, 9 October 2025 (UTC)
:::: It may give you that right from the ''copyright'' perspective, but perhaps not from ''academic publishing integrity'' perspective. Unfortunately, I do not have any guideline handy; I am merely following my common (or not so common) sense. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:32, 9 October 2025 (UTC)
:: I would like to ask: was this article guided by someone from an academic institution, such as a university? Is it reviewed at least in some weak sense? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:39, 9 October 2025 (UTC)
:::Yes, This article has been reviewed by two academic professors, their names are also listed as co authors.
:::First, a project guide would help us with selecting a topic and with the document 
:::Second, an Internal examiner would go through our experiment and approve it
:::Finally, External Examiner would examine the documentation and verify it. 
:::We were required by these professors to put their name under contributions [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 05:48, 9 October 2025 (UTC)
:: Let me explicate the promotional potential of such a page a bit: one can go to the page of the article in Wikiversity --&gt; https://tomjoejames.com/ --&gt; HitMyTarget (a commercial, profit-making entity?) Why would the link be to a commercial web site rather than an academic page, or perhaps a LinkedIn account, which I think the person has? There could also be no link at all; a search for the name would turn out something in Google as well. But providing a direct link would drive users/viewers toward that website much stronger since otherwise the viewer of the page would have to open a new Google search window or the like. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:45, 9 October 2025 (UTC)
:::It is evident that the website is not even close to being complete. 
:::I will be creating a separate page under the same domain name specifically for people to contact me.
:::The url would probably be defined as tomjoejames.com/contact-me/
:::I haven't decided yet. But that is my personal website.
:::If the community requires me to remove it, I will. But personally I think people who are from here most likely to click the link to know more about me or to contact me. Either way I think my personal website serves the purpose. 
:::As for the HitMyTarget, it can be traced from any of my links. From my research gate profile, linkedin page or even my own userpage.
:::On the article I did not add any promotional content about myself, I hyperlinked only my own name. I do not know how that is promotional. [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 06:04, 9 October 2025 (UTC)
:::: I am pausing any further responses from me to see whether anyone else has any input. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:30, 9 October 2025 (UTC)
:What does it mean &quot;There is perhaps no more appearance/suspicion of copyright violation&quot;? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:57, 16 October 2025 (UTC)
:I have accepted VRT permission per [[ticket:2025100410001149]] FYI. [[User:Matrix|Matrix]] ([[User talk:Matrix|discuss]] • [[Special:Contributions/Matrix|contribs]]) 11:00, 28 October 2025 (UTC)
::Thank you Matrix [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 12:43, 28 October 2025 (UTC)
:I would '''delete''' it. 1) it states its a learning resource. It could not be a learning resource as not rewieved original research. 2) It is not an ongoing research, nor the research was performed on Wikiversity - wv is not a preprint or article database. Maybe it could be moved elsewhere withn Wikimedia domain, but I dont know where. So I would delete it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:56, 20 November 2025 (UTC)
::I would '''keep it.''' Like Dan had pointed out,  we do have article-like pages in Wikiversity, and this is not just a random pseudo science article but an article that is a report of an final year project, it has been reviewed by 3 professors whose name has been mentioned at the very beginning.   [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 14:50, 21 November 2025 (UTC)
:::I think it is not good to rate pages by appearance. It can be done on other Wikimedia projects, but it cannot be done on Wikiversity, because Wikiversity does not create a static format for presenting information, but is focused on the goal and process. Unfortunately, the goal and process do not have a uniform format. While a target article on Wikipedia or an entry on Wiktionary have some standard target format, Wikiversity does not. That is why I personally rate pages according to the goals and their assessment. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:05, 22 November 2025 (UTC)

Further reading for this nomination: [[S: Wikisource:Proposed_deletions/Archives/2025#Index:Cookie_Encryption.pdf]]; EncycloPetey handled the matter. Let me quote his wisdom on Zenodo (which I lack): &quot;This is tied to a PDF on Commons that was uploaded as &quot;own work&quot; with a CC license and a doi link to Zenodo, with no indication of where this paper was published or if it was published. Zenodo is not a publisher; it is a site for storing research and sharing papers. If Zenodo is the only place this was &quot;published&quot; then it was effectively self-published. --EncycloPetey (talk) 16:14, 15 September 2025 (UTC)&quot;

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:55, 9 October 2025 (UTC)

:Can you clarify what point are you trying to state? Didn't I already state that the article is published by me? 
:I first created the article in wikisource which I thought would be the perfect place, unfortunately they do not allow self published articles that are not notable. Then I discovered Wikiversity where they allow self published articles. That is why I created the article here. 
:Unlike in wikisource, I did follow guidelines.
:Ever since you deleted the first article, I spent time reading Wikiversity guidelines and I do think that I am following it perfectly.
:I would like to get your suggestions on how should I improve the page,  10 points would be sufficient. 
:Because your goals or intentions are confusing me very much. At first you told me that the article is exactly the same as the preprint in RG and therefore there is no use to it here. And then when I continued to optimize it for Wikiversity, you went ahead and said it is problematic according to recommended academic publishing. 
:Atleast just respond to the points that I have made whether you agree or disagree. So that I clarify and proceed to discuss points that are important and relevant 
:Have you published an research article? If yes, could you send it to me so that I can see the format you have done it [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 10:45, 9 October 2025 (UTC)
:: I am giving a chance/time to other curators/custodians to look at the matter and respond to my inputs. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:14, 9 October 2025 (UTC)
:: Incidentally, above I counted 4 questions (or more), 1 request (or more?) and 1 command (or more?). That is a behavior of a commanding entity. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:24, 9 October 2025 (UTC)

I would '''delete it''''. It's more like an academic communication than a learning resource or research.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:32, 26 October 2025 (UTC)
:: In the above post, I do not see any valid rationale for deletion: we do have article-like pages, in Wikijournals and also e.g. in [[Physics/Essays/Fedosin/Stellar Stefan–Boltzmann constant]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:59, 3 November 2025 (UTC)
:::But I do, see above. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:56, 20 November 2025 (UTC)

:it is a '''student research paper''' forming part of a learning resource on web security and encryption.
:The project was conducted as part of a final-year university course and documented as a practical study on cookie encryption and it has been reviewed by three professors. However, I will be creating a sub page for the article to elaborately describe the experiment that we have conducted and the results we got.    [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 15:57, 26 October 2025 (UTC)
::And why should w host research papers? Wikiversity is not an academic Journal nor repository. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:06, 22 November 2025 (UTC)
:::I do not wish to go through this same argument once again, I've already answered to this question several times in Dan's talk page, Colloquium. you can refer them. I am not hosting the research paper here, I have already hosted the pdf in the ResearchGate, I have published a text version in the wikiversity so that it may be useful for others. Unless you can show me how that article is totally useless, I would like to '''keep''' the article in the wikiversity.    [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 10:13, 22 November 2025 (UTC)
::::And thats the point I am having. Wikiversity is not paper repository. The only way is to publish it via WikiJournal, but they want it for Wikipedia usually. Why wikiversity should be a duplication of ResearchGate, Academia or Zenodo?
::::What I can read on [[Wikiversity:What is Wikiversity?]] policy is, that Wikiversity research &quot;...includes interpreting primary sources, forming ideas, or taking observations.&quot; The article doent look to fall into this. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:43, 22 November 2025 (UTC)
:::::Well, then how come you missed the term &quot;Learning Projects&quot;? As Jtneill had pointed out, this is a legitimate learning project. And also, I do have the VRT permission to host this article on Wikiversity.  [[ticket:2025100410001149]] . besides ResearchGate is an self-archiving platform. the document version in it is not accessibly to screen readers (usually disable people use them), Translators, and also for the mobile readers. therefore I do have valid reasons to publish this article on wikiversity. 
:::::# It is a learning project, therefore according to WIkiversity Policy, It qualifies. 
:::::# I have an explicit VRT permission to host this article on Wikiversity
:::::# Versions that are published in RG, Zenodo are documents, and they are not accessible by screen readers or mobile users. Therefore it is imperative that an article version of this paper exist on here. 
:::::Therefore this article qualifies to stay here on Wikiversity.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 11:22, 22 November 2025 (UTC)

'''Keep'''. This is a legitimate student learning project that may be of use to others. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:51, 22 November 2025 (UTC)
{{archive bottom}}

== [[Pragmatics/History]] ==
{{archive top|Deleted. Other related resources have been deleted. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:24, 17 May 2026 (UTC)}}
Another KYPark page and subpages with unclear organization scheme. Contains fairly many redlinked items. See also [[User:KYPark/Literature]], perhaps a similar concept. Unlikely to be really useful for others but KYPark. '''Move to user space'''.

As an alternative, moving to [[History of Pragmatics (KYPark)]] would make sense to me: the topic is identified using a natural-language phrase (instead of the relatively unnatural slash) and the responsible editor is indicated so that the reader knows whether to look or not. And for those who oppose the brackets (which I like): [[History of Pragmatics/KYPark]]. Or also: [[KYPark/History of Pragmatics]]. But then, searches in mainspace will see that content and the question is whether that is good. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:21, 15 October 2025 (UTC)

:What about to propose the user to write some guidelines, how other can participate instead of deleting it? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:03, 16 October 2025 (UTC)
:: I plan to move the pages to userspace as I proposed. If someone wants to ask KYPark to address the problems, they should feel free. There will be plenty of time for KYPark to address the problems while the material is in user space. After the problems are addressed, the material can be moved back to mainspace. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:38, 15 November 2025 (UTC)
:So I would '''delete''' it. In the blocked user space its useless. The user cannot improve it and Wikiversity is not free hosting service for personal pages. My believe is, that there should be just a few working pages in the users spaces. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:30, 11 March 2026 (UTC)

'''Move'''. Insufficient statement of learning objective or connection to related learning resources with insufficient current activity to stay in main space. The page was originally [[History of pragmatics]] but was moved by Dave B. Therefore, I suggest moving to [[User:KYPark/History of pragmatics]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:57, 22 November 2025 (UTC)
{{archive bottom}}

== [[IMHA Research Archives]] ==
I propose to '''move to userspace''', including the subpages. I struggle to understand how Wikiversity readers are supposed to benefit from the material here and in the subpages. In the log, there is e.g. '10 February 2019 Marshallsumter discuss contribs deleted page IMHA Research Archives (content was: &quot;{&lt;nowiki/&gt;{Delete|Author request}} Thanks! -&quot;)', so the page was deleted before, but not the subpages.

We could also delete all the material if we have strong enough suspicion too much of it is copyright violation. In any case, moving to user space improves the matter a little by moving the content away from Google search. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:38, 9 November 2025 (UTC)

:Looking at some sub-pages, they can be deleted e.g., because they only consist of broken links or are largely empty. I deleted a couple but haven't been through all to check. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:27, 10 November 2025 (UTC)

As an example, let me give the wikitext content of [[IMHA Research Archives/3. Scientific litterature search, storage and use]]:
&lt;pre&gt;
==[[/Medicina Maritima - the Spanish scientific maritime health journal/]]==

==[[/PubMed/]]==

==[[/Google and Google Scholar/]]==

==[[/Zotero/]]==

==[https://www.dropbox.com/sh/d91z7bcyelfvk42/AAAkIvjtBnnFMbiU9ZLOdVL9a/Andrioti_database%20sources0310.pptx?dl=0 Maritime health web portal ressources ]==
&lt;/pre&gt;
The wikilinks are red; the external link to dropbox says &quot;You don't have access&quot;. This was made in 2016. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:04, 11 November 2025 (UTC)

:I suggest delete -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:27, 12 November 2025 (UTC)
:: I think we should avoid deletion as much as possible, instead moving to user space (bar copyvio, ethics violation, etc.). This is a good general principle. It greatly improves auditability and makes it so much easier for anyone to request undeletion since they know what content they are requesting for undeletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:52, 12 November 2025 (UTC)
:::Do not recreate Wikiversity from the educational and research project to the personal blog. That will lead to the cancelation of it by WMF. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:44, 20 November 2025 (UTC)
:::: The English Wikiversity has a long tradition of moving problematic content to user space, as per evidence collected at [[User:Dan_Polansky/About Wikiversity#Moving pages to userspace]]. If Wikimedia Foundation finds this problematic, they can start a discussion in Colloquium and state their concerns. They do not need to make explicit threats at first; they can start a discussion and explain why it is problematic. They can even do it from an anonymous IP and provide a well-articulated reasoning. And anyone else can start a discussion in Colloquium to change this tradition. I do not see why we should not want to change that tradition based on well-articulated, compelling reasoning. I see no reason why Juandev should be making threats instead of them, on a per RFD basis. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:58, 21 November 2025 (UTC)
:::: If Juandev is ''sincere'' about deleting very-low-value items ''from user space'', he should perhaps demonstrate that by asking his pages like [[:cs:Uživatel:Juandev/Problémy/Kov/Repase dvířek elektroskříně]] to be deleted; otherwise, I register a ''glaring inconsistence''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:43, 21 November 2025 (UTC)
::What was the original delate page about @[[User:Jtneill|Jtneill]]? I guess that would be crucial for the decission. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:48, 20 November 2025 (UTC)
:::@[[User:Juandev|Juandev]] the couple of pages I checked and deleted were much like @[[User:Dan Polansky|Dan Polansky]] posted above i.e., headings with empty sections and/or broken links but no substantive content. But I think each sub-page needs checking. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:59, 20 November 2025 (UTC)
::::So I'm saying that the main page usually determines what the other pages are for. But if I don't know the page because it's been deleted, or why was deleted (deletion based on the founder's request is probably not the rule), it's hard to judge. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 22:16, 20 November 2025 (UTC)
:::::I've pasted the original content of the root page: [[IMHA Research Archives#Original page]] (i.e., prior to the content being removed and deletion requested) to help understand the context for the sub-pages. In 2018, Saltrabook blanked the page, indicating that the content had been moved elsewhere, and requested page deletion. Marshallsumter then deleted the main page but not the sub-pages. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:58, 21 November 2025 (UTC)
::::::I see, so if those subpages are usefull I would keept them, if not I would delete them. I dont see a point of providing free hosting to sombody, by moving many pages to their user space. The question is if we want to host (i.e. to have in the main ns) lists of links elsewhere. I have no opinion on that. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:11, 22 November 2025 (UTC)

: Let me clarify that while many of the subpages are like the example above, [[IMHA Research Archives/Scientific litterature search, storage and use/Zotero]] is different:
:: &quot;A continuous critical and evidence based learning is a core issue in clinical practice, research, teaching, publication and prevention activities. The Zotero Program is just one of many scientific literature management programs, that should be used for these purposes. Of course one can live without such a database but it helps a lot and can save a lot of time that could be used for more interesting issues. Not only that, but it helps to create better publications and knowledge. Without this program it can be very time consuming to publish a scientific article with the requested style for the references. Further in daily practice when you want to collect and cite a few references for a specific evidence in a clinical colloquium and discussion, this program is excellent. Therefore we strongly recommend that all maritime health persons learn how to use this excellent tool in their daily maritime health practice of all different types. There are good online courses for self-instruction on how to use Zotero. For example this one: Zotero fast online course But in order to increase IMHAR´s collective scientific strength in the use of EBM we would like to give training sessions in every possible opportunity, IMHA Symposia, seminars and other types of meetings. The database is useful for personal purposes but especially also for collaborative aims. At the IMHAR meeting in Paris Oct 7th 2016 we will give an introduction to the program by showing how it can be used in the daily practice and discuss strength and weaknesses compared to other similar databases.&quot;
: Even longer is e.g. [[IMHA Research Archives/Scientific litterature search, storage and use/Medicina Maritima - the Spanish scientific maritime health journal]].
: However, that does not mean these should be salvaged. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:53, 21 November 2025 (UTC)
:{{ping|Saltrabook}} I'm wondering if you can respond here to help us decide about whether to delete the IMHA Research Archives sub-pages or perhaps move them to your user space? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:58, 17 May 2026 (UTC)

== [[Palliative medicine]] ==
{{archive top|'''Kept'''. Page has been improved. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:38, 17 May 2026 (UTC)}}

Underdeveloped and has not been improved on since 2007. Author inactive. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:42, 14 December 2025 (UTC)

:Delete, per nominator [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:16, 22 January 2026 (UTC)
:Yes, I would also expect there to be more and especially that someone would write how to use it. However, it still seems to me to be a useful thing in the sense of a syllabus, so that someone who is interested in the topic knows what information to obtain in order to get a complete picture of the topic. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:55, 16 March 2026 (UTC)
{{archive bottom}}

== [[Canadian Wilderness]] ==
{{archive top|Deleted per nom. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:07, 17 May 2026 (UTC)}}
This page doesn't seem to belong to wikiversity. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 09:55, 6 February 2026 (UTC)

:In principle there could be some material useful here but in practice, I don't see what this page is adding as an educational resource. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 12:54, 6 February 2026 (UTC)
:I can see this being a useful resource to a bigger project. Maybe we could move it to the &quot;[[Wikiversity:Drafts|Draft]]&quot; namespace vs. deleting it? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:28, 6 February 2026 (UTC)
::Does anyone plan to work on it? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 01:59, 8 February 2026 (UTC)
:::Next week the page has it's 17th birthday. Ever now and than someone added to it. With a lot of work it could be a nice encyclopedic article but making it educational .... Merging it may take more work than rewriting it. Move to Draft might be the best option. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 08:58, 12 February 2026 (UTC)
{{archive bottom}}

== [[LQR Control for an Inverted Pendulum]] ==
{{archive top|'''Deleted''' per nom. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:36, 17 May 2026 (UTC)}}
Underdeveloped resource, has not been edited for more than a decade. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:03, 16 March 2026 (UTC)

:Looks like a test, '''delete'''. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:30, 27 March 2026 (UTC)
{{archive bottom}}

== False flag &quot;authority hack&quot; user page deletion ==
{{archive top|'''Not undeleted''', the requester dropped the request. See Wikiversity:Requests for Deletion v. 2803217.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:36, 10 May 2026 (UTC)}}
'''Undeletion requested'''

Hi, Juandev marked my user page as &quot;spam&quot; and &quot;authority hack&quot;, and deleted it. 

First, I asked him for help with &quot;time limit for new users&quot;, and he replied - I should admit I dont know, what is &quot;new user limit&quot;, but if filter blocks your page because of certain external link, you may force to save anyway and sometimes it works. It should not work, when the website is blacklisted. As of now, I am not seeing you to save page in main namespace, so try to save it without external links first. 

Then he wrote me another message: Well, I have analyzed your contribution to Wikiversity and I should point out here, that this project is not a place for advertising, so there is no way of promoting your books and authority this way. - probably referring to the intro of my About me page where I present me and my work. 

Before I could explain him the difference between the neutral information and advertising and promotion, he deleted my user page. 

Here is my answer I posted to the discussion today: 

: Hi, my About Me page is just an info page with the neutral as possible presentation of my work.
:
: There is a big difference between informing and advertising. Informing is neutrally stating that something exists and requiring no action, while advertising is a special communication form with intent to cause certain action from readers. For example, click here, click there, order this, buy that.
:
: There is no such intention, form, or terms on my info page. Just neutral information. I don't hide and I am not ashamed that I am write and author, and that is a part of the usual bio, including works. I checked your user page: &quot;I graduated from the Czech University of Life Sciences in Prague and studied information science at the Faculty of Arts of Charles University.&quot; I think that if you had written a book on Life Science, you would have mentioned that as well.
:
: Most of the Info page is about my research and AIPA Method which is a valid contribution to psychology, consciousness studies, identity theory, and personality development. Actually, my paper '''AIPA Method: A Cognitive-Phenomenological Model for Identity Reconstruction and Stabilization in Pure Awareness''' is now in the peer review procedure at Journal of Consciousness Studies.
:
: Here is a part from the Wikiversity AIPA Method page in creation (waiting for the end of the time limit for new users):

: == Introduction ==
: The AIPA Method addresses a gap in contemporary personal development and consciousness science: most evidence‑based approaches (CBT, MBSR, MBCT, standard meditation) operate at the level of mental content—reframing thoughts, observing them, or reducing their impact—rather than at the level of identity structure. In contrast, AIPA targets the structural relationship between the self and the mind, aiming at durable identity reconstruction rooted in Pure Awareness rather than symptom management.
:
: The central research question of the primary AIPA preprint is whether a structured, sequentially staged method can produce permanent identity reconstruction rooted in Pure Awareness, and how such a method compares to established approaches in scope, mechanism, and outcome.
:
: == Theoretical foundations ==
: The AIPA framework is grounded in the cognitive‑phenomenological tradition (e.g., McAdams, Varela, Metzinger, Erikson), contemporary consciousness science on minimal phenomenal experience, and qualitative methods advocacy in psychology. It builds directly on:
:* Empirical work on pure awareness and Minimal Phenomenal Experience (MPE), especially Gamma &amp; Metzinger’s large‑scale study of content‑reduced awareness states.
:* Metzinger’s proposal of minimal phenomenal experience as an entry point for a minimal unifying model of consciousness.
:* Narrative identity and partial‑self models within personality and identity theory.
: Within this backdrop, AIPA proposes Pure Awareness as a distinct, operationally specified state that can become a structural ground of identity rather than a transient meditative experience.
:
: == Experiential empiricism ==
: The empirical foundation of the AIPA Method is explicitly first‑person and experiential, combining:
:* A 22‑year longitudinal autoethnographic self‑study (2003–2025) documenting partial personality episodes, protocol use, and outcomes.
:* A 13‑year prospective verification period with zero self‑reported recurrence of targeted harmful behaviors after a dated stabilization point (1 January 2006).
:* A high‑ecological‑validity “stress test” during acute bereavement, used to examine whether non‑reactive awareness remains stable under maximal provocation.
:* Two independent practitioner cases (an Amazon‑verified report and a structured questionnaire case) providing preliminary convergent signals across cognitive, emotional, behavioral, and identity dimensions.
:
: All central constructs (Pure Awareness, partial personalities, the Switch, identity stabilization) are operationalized with explicit phenomenological and behavioral criteria intended to enable replication and future third‑person measurement.
:
: I believe this is a valid contribution to Wikiversity.
:
: Best regards, Senad [[User:Senad Dizdarević|Senad Dizdarević]]


I suggest you check the deleted user page, and see for yourself if it is &quot;spam&quot; and &quot;authority hack&quot;, or a legit author's page with one paragraph short presentation, while the rest of the page is about my research project. 

Thank you for undeleting my user page, so I can use it. 

Best regards, 

Senad Dizdarević [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:26, 2 April 2026 (UTC)

:Hi Senad,
:Welcome to Wikiversity.
:It looks like you tried adding similar content to Wikipedia and ran into similar difficulties over there ([[w:User talk:Senad Dizdarević]])? Perhaps that is what has led to you Wikiversity?
:Basically, if you'd like to collaborate and help build open educational resources, feel free to contribute to Wikiversity. But if the primary motivation is to promote your autobiographical work you're probably going to run into challenges.
:Sincerely,
:James
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:11, 3 April 2026 (UTC)
::James, Hi, and thank you for your answer. 
::Yes, in 2025, I created the autobiographic page on Wikipedia, which was removed because of the links to my books on Amazon. To admin, I explained that I did not know the rules, and agreed that page is removed. Now I know that somebody else must write a Wikipedia page for you. 
::On the deleted user page on Wikiversity, there were no links to Amazon or any other form of promotion, just neutral as possible basic presentation of my writing (one sentence) and current project (the rest of the page). 
::I created Wikiversity page to present my AIPA Method project, to invite researchers to read it, give their opinion, and conduct empirical researches in their institutions. Now, it is in a theoretical phase, and needs more empirical testing. 
::Best regards, 
::Senad [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:03, 3 April 2026 (UTC)
:::It looks to me like the primary motivation for contributing to Wikiversity is to drive traffic / search engine ranking to your website?
:::* [[User:Senad Dizdarević]]
:::* [[AIPA Method]]
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:36, 4 April 2026 (UTC)
::::No, it is not. There is no link to my website, so &quot;driving traffic to my website&quot; is not possible. 
::::For your educational purposes: 
::::Copilot &quot;AI:  [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:38, 4 April 2026 (UTC)
:::::So do you still insist of undeleting your former version of your userpage if you have created the new one? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:15, 6 April 2026 (UTC)
::::::No, because in the moment of undeletition, somebody could delete it again, and so on. Thank you for not deleting my new user page, as it is made in your user page image.  [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 08:59, 6 April 2026 (UTC)

{{archive bottom}}

== Undeletion request ==

It was deleted by an admin without discussion and with untrue rationale. If people take offense with the question that doesn't mean it's not a valid question and the page was good. Please undelete the Wikidebate page [https://web.archive.org/web/20250810030352/https://en.wikiversity.org/wiki/Is_it_likely_that_Earth_has_been_visited_by_aliens_millions_of_years_ago%3F Is it likely that Earth has been visited by aliens millions of years ago?]

There are lots of sources on the subject, the wikidebate is sourced very well compared to other wikidebates and wikiversity pages, and the page is educational, useful and of good quality. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 23:57, 10 April 2026 (UTC)

:Page: [[Is it likely that Earth has been visited by aliens millions of years ago?]]
:Ping: [[User:Atcovi|Atcovi]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:21, 11 April 2026 (UTC)
:There is no need for a discussion for straight garbage-level, pseudoscientific content.
:For '''Is it likely that Earth has been visited by aliens millions of years ago?''', the flaws for this page wouldn't even take someone more than a few minutes to assess:
:* Essentially, the &quot;pro&quot; arguments unproven claims being derived from irrelevant, established facts (basically: &quot;it is likely aliens have came because Earth has existed for so long [sources proving Earth's longevity]&quot;). These are not serious, scientifically-backed arguments - these are non sequiturs. It's as if I said Wikipedia has existed longer than my existence on Earth ([https://d1wqtxts1xzle7.cloudfront.net/74351725/eyJoIjogImZiODhmYzNkODU1N2UxMWExYzUyODJiYzgzZTRmZDM4OTBjODY5YWMzMjA3NDNmOWEyZTA0ZTU3ZGYwZjAyYTkiLCAidSI6ICJodHRwczovL3B1cmUuaHZhLm5sL3dz-libre.pdf?1636354596=&amp;response-content-disposition=inline%3B+filename%3DCritical_Point_of_View_A_Wikipedia_Reade.pdf&amp;Expires=1775872055&amp;Signature=GqbUZboYRvUYWi~aW40LT5eZSHrLuDL3o0-DxAH8vSvcJcGAuyByZWLF2oHTY6GlB72TqvZxpE-v9d4gvsA6myriYqO~QQQZgWxjT2JXjUWC-yiPcTF4l~lroJSi4dY0v9eKiBcU03l-aeUdrX8~UPfi0TfW0IhsmzH-VBR6X6FrzRpIqc6uM6n9YXfr5FRB3aCqqokU690af3n0Hguaub1Zgmh9qjYYqzBS0VOOHjKTTEQnDuadX3jl5CQeXYTaeCC3H0hMeVwHlratbrnuFEKC1aN0-5znCUoSzMEg21ECzGPTrSDM1W05dcK-u0ZTCeUGKAuC-2yRFL3sY46MIw__&amp;Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA#page=157 reputable source proving ''this'' fact]), therefore it's likely that my birth took place solely for the sake of me experiencing Wikipedia (0 backing). It makes no sense and no person with at least a high school-level of intelligence would take this seriously. 
:* What is worse is that the user is being misleading with their &quot;[the page is] sourced very well&quot; claim. The sources ''themselves'' don't even back up the claims. It's just used as proof for an established concept, where the user then uses this established concept to jump to an unsupported, laughable conclusion that is pulled out of thin air. It's utterly ridiculous to even consider such a page for mainspace since it clearly violates our [[Wikiversity:Verifiability]] policy. This is, once again, pseudoscientific content that has caused our website to reduce in quality over the last few years. 
:* Going source by source, we can see that:
:#[https://web.archive.org/web/20250918011642/https://timesofindia.indiatimes.com/blogs/thebigd/compress-earths-history-into-24-hrs-humans-came-at-1158-pm-yet-killed-70-of-wildlife/ ‘Compress Earth’s history into 24 hrs. Humans came at 11:58 pm, yet killed 70% of wildlife’] is literally just a blog post which doesn't even mention aliens or extraterrestrial life. It just talks about Earth's history in accordance with the 24-hour metric of time, and the author tries to use this article as a 'piece in the puzzle' of aliens &quot;possibly&quot; visiting Earth... which, once again, is unsupported and is not backed up anywhere in the article.
:#[https://web.archive.org/web/20250808053249/https://news.cornell.edu/stories/2023/11/jurassic-worlds-might-be-easier-spot-modern-earth The Cornell article does not even remotely support the idea that &quot;aliens visited Earth&quot;]. It mentions a ''chance'' of &quot;life there [a habitable exoplanet] might not be limited to microbes, but could include creatures as large and varied as the megalosauruses or microraptors that once roamed Earth.&quot;, but again, no justification to take this article as proof that &quot;aliens may have visited us!&quot;. There's no mention of aliens visiting Earth anywhere in the article. Once again this is only proving the background premise, but not the unsupported, nonsensical &quot;alien likelihood&quot; argument that the author of this garbage page is trying to push so desperately.
:#The Parker Solar Probe WP article does not even mention aliens either. It follows the same issues as the previous argument.
:And the other page this user complained about [https://en.wikiversity.org/wiki/User_talk:Atcovi#Deletion_of_educational_page_because_of_personal_opinion on my talk page] holds almost similar, maybe even more fatal mistakes, than this one. It has nothing to do with &quot;taking offense&quot;, this is just low-quality, garbage content. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:56, 11 April 2026 (UTC)
::Why do you think pro claims are required to be proven? It's possible to object to them and these are arguments, not contextualized to be statements of proven facts. And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future. Instead of insulting others' intelligence, maybe engage with the actual reasoning rather than censoring it away. And there are lots of sources, such as [https://interestingengineering.com/science/alien-civilizations-may-have-visited-earth-millions-of-years-ago-study-says Alien Civilizations May Have Visited Earth Millions of Years Ago, Study Says] etc etc. The sources are used for the arguments themselves individually. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 12:30, 11 April 2026 (UTC)
:::Because, once again, this is not a site that caters to rampant debating for the sake of &quot;we need to employ rationality and logic to solve the world's problems&quot;, we have policies that we need to fulfill. The claims made in the pro argument clearly do not meet [[Wikiversity:Verifiability]], since you cannot verify these arguments with the sources because they are not relevant. 
:::''&quot;And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future.&quot;'' The point being is that these arguments are not supported by the sources. Even the article you mention poses the idea as a hypothetical model. This is just you twisting the article to fit your unsupported narrative. I'll bring direct quotes for you to show why the linked article does not help you:
:::* ''One problem the researchers do make sure to point out is that '''they are working with only one data point: our own behaviors and capabilities for space exploration'''. “We tried to come up with a model that would involve the fewest assumptions about sociology that we could,” Carroll-Nellenback told Business Insider. '''We have no real way of knowing the motivations of an alien civilization'''.'' --&gt; proves that this is just speculation and no evidence-based arguments have been provided for the idea that aliens likely visited Earth.
:::And I'm not sure if you read my entire response, but I ''did'' engage with your &quot;actual reasoning&quot; and exposed its weaknesses and lack of adherence to Wikiversity policies. If we allowed content that was just filled with non sequiturs we would have content that fails Wikiversity's educational objectives and reduces the overall quality of this website, hence why such a harsh stance needs to be taken. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:50, 11 April 2026 (UTC)
::::Thanks for proving that the Wikimedia ecosystem is unfit to deliberate on controversial topics. The question is entirely valid and the content is far better sourced than nearly all Wikidebates and has no genuine flaws. The only possible issue with it as far as I can see is that now that Wikidebates has been paused people can't add objections if they do have sth specific to say about the topic that's not already included on that page which already had plenty of Cons and objections.
::::The page was more educational than most of Wikiversity and it was well-sourced – wikidebates was for arguments so people were invited to make arguments based on sourced things or outlined logic and the page met [[WV:V]] and most pages on Wikiversity aren't sourced as good. Doesn't look like people can see beyond their biases and personal views here but that's more evident in the marginalization and deletion of wikidebates and the low activity in that project than these selective deletions. A constructive thing to do would be to add reasoned Cons and objections not yet on the page and people had plenty of time to do that. There are and will be other sites for free constructive rational adversarial deliberation (not a big loss in that sense). [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 16:31, 22 April 2026 (UTC)
:::::Thank you for failing to address any of my arguments and going on an unrelated, nonsensical tangent that has nothing to do with the discussion. Once you start producing work that aligns with Wikiversity's content policies instead of typing up laughable, pseudoscientific garbage, then maybe your work can be accepted and not removed. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:59, 22 April 2026 (UTC)
::::::I suggest you stop ridiculing things and learn respectfully forming genuine points about the subject at hand. {{tq|the idea as a hypothetical model}} but please learn first about what arguments are and why they're not the same as a statement of objective proven fact. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 17:18, 22 April 2026 (UTC)

==Pages by Harold Foppele==
[[User:Harold Foppele]] is locally blocked indefinitely and globally banned for sockpuppetry. There were also WMF and local community concerns expressed about copyright violation and AI (over)use. As a result, I think the Wikiversity pages created by this account warrant review with regard what should be deleted, what should be retained etc.:
* [[Completing the square]]
* [[Number of independent spatial modes in a spherical volume]]
* [[Quantum]]
** [[Quantum/Andrew N. Jordan]]
* [[Quantum A Matter Of Size]]
* [[Quantum A Spooky Action at a Distance]]
* [[Quantum: A Walk Through the Universe]]
* [[Quantum Computing Algorithms in the NISQ Era]]
* [[Quantum Formulas Collection]]
* [[Quantum harmonic oscillator]]
* [[Quantum Matter Elements and Particles]]
* [[Quantum mechanics]]
** [[Quantum mechanics/Timeline]]
* [[Quantum mechanics learning module]]
* [[Quantum mechanics measurements]]
* [[Quantum Noisy Qubits]]
* [[Quantum optics beam splitter experiments]]
* [[Quantum: The Secret of Cohesion: How Waves Hold Matter Together]]
* [[Quantum Ultra fast lasers]]
* [[Speed of sound experiments]]
* [[User:Harold Foppele]]
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:12, 17 May 2026 (UTC)

:'''Delete all''' Not worth keeping. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:27, 17 May 2026 (UTC)

== [[Classical guitar pedagogy]] ==

According to the talk page, the author of this page intended to create this page for Wikipedia. At this moment in time (nearly 20 years later), the page is still riddled with red links and doesn't seem to fit Wikiversity's learning modules. Therefore, I propose that this page should be deleted. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:03, 19 May 2026 (UTC)

== [[Film writing]] ==

Undeveloped since 2007. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:05, 19 May 2026 (UTC)</text>
      <sha1>66gignailrf8meb1tad97kedqn8ew52</sha1>
    </revision>
    <revision>
      <id>2810516</id>
      <parentid>2810422</parentid>
      <timestamp>2026-05-20T00:25:22Z</timestamp>
      <contributor>
        <username>Koavf</username>
        <id>147</id>
      </contributor>
      <comment>/* Classical guitar pedagogy */ Reply</comment>
      <origin>2810516</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="58331" sha1="i2ge7u3jyfcgxnler4n8flhn6iee08q" xml:space="preserve">{{/header}}
== [[Korean/Words]] ==
(I go to RfD instead of ''proposed deletion'' since many pages are affected.)

I proposed to quasi-delete, i.e. '''move to userspace''' of the main (or sole?) creator, {{User|KYPark}}.

The page is organized a little bit like a dictionary. It makes it redundant to Wiktionary except that Wikiversity allows original research and there does seem to be original research there. Thus, its being organized as a dictionary would alone not necessarily be a problem.

Where I see a problem is in the organization and execution/implementation. Consider [[Korean/Words/가다]], which seems rather typical of the subpages (some subpages are like categories and transclude the pages for individual words):
* On the putative definition line, there is this: &quot;한곳에서 다른 곳으로 장소를 이동하다&quot;, apparently(?) in Korean. That does not seem to fit well into the ''English'' Wikiversity.
* There seems to be some original research into etymological relations between Korean and European languages in the &quot;Comparatives&quot; section (from what I recall, the English Wiktionary rejected this kind of content from KYPark). Admittedly, it is marked using &quot;This is a primary, secondary and/or original Eurasiatic research project at Wikiversity&quot;, so it could be tolerable, but even so, one has to wonder whether Wikiversity wants this kind of fringe science/research or outright pseudo-science.
** Fringe science: fringe physics has been moved to user space before. This would be fringe etymology. But then, original research is allowed.

Deletion is not required; moving to user space suffices, I think. Alternatively, one could at least rename the pages to make it clear from the title that this is not Wikiversity voice but rather KYPark voice, e.g. &quot;Korean/Words (KYPark)/...&quot; or  &quot;Korean/Words/KYPark/...&quot;  (recall the &quot;Fedosin&quot; pages featuring the name &quot;Fedosin&quot;).

Methodology: I see almost no methodological notes spanning the words at [[Korean/Words]]. And yet, if this is original research inventing new etymological connections, surely there should be some general considerations/analysis on how to proceed and how that manner of procedure differs from mainstream etymology?

Prefix index (max 200 items?):
{{collapse top}}
{{Small START}}
{{Special:Prefixindex/Korean/Words}}
{{Small END}}
{{Collapse bottom}}
--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:33, 24 September 2025 (UTC)

:I would keep it. If there is a course of Korean, why not to have a resesearch on Korean vocabulary? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:53, 16 October 2025 (UTC)
:: I propose to dismiss the above input: 1) it does not contain any argument, except for a question, and a question is not an argument (it can be so reinterpreted, but that includes additional burden on the interpreters, in interpreting it the wrong way); 2) it ignores all the issues I have raised, including that there is something like definition lines in Korean, in this ''English'' Wikiversity. To answer the question asked: there can be a research on Korean vocabulary in the mainspace, but not one showing the defects I identified above. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:35, 15 November 2025 (UTC)
:I've reviewed a sample of approximately 20 of the Korean/Words sub-pages and lean towards moving to user space because:
:* The pages appear to be an idiosynchratic collection of etymological pages about Korean language
:* There is minimal English instruction which is problematic for English Wikiversity
:* There is no explanation of research method
:* There is no educational rationale
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:31, 22 November 2025 (UTC)
:Well, since the original creator has indef I change my mind and I would '''delete''' it. The case is nobody knows how to continue with the research and if we move it to the userspace, the user cannot improve it eihter. What the original user can do to request admin, to send them a contentent to their email for example if they really want to improve the resource elsewhere. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:38, 11 March 2026 (UTC)

I think the consensus here is delete. {{U|Codename Noreste}} do you know an efficient way to mass delete these pages? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 17 May 2026 (UTC)
: I would use a script, but I would probably not delete those pages yet until we have the pseudobot user group. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:53, 17 May 2026 (UTC)

== [[Enhancing Web Browser Security through Cookie Encryption]] ==
{{archive top|'''Kept'''. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 17 May 2026 (UTC)}}
To avoid further conflict with the user who entered this text into Wikiversity, I am opening a RFD request.

I am not sure about how to proceed, although I am inclined to move it out of mainspace = quasi-delete. I am looking forward to get input from others, especially curators and custodians. Some considerations:

1) There is perhaps no more appearance/suspicion of copyright violation, now that the ResearchGate (RG) article (of which this is a copy, perhaps an incomplete copy?) carries a license.

2) The article is not a complete replica from RG: at a minimum, it lacks images. The inserter could have continued editing the page in his user space before he uploads images, that is, before he finalizes the page for consumption, but that did not happen. I did not check whether the text is an exact one-to-one match; the article does not indicate anything in that regard.

3) The principle implied seems to be this: users should feel free to duplicate non-peer-reviewed articles from RG in English Wikiversity, perhaps to increase the Google search and LLM yield. I find this problematic, in part for the duplication. I would say: choose a venue and publish it there. If RG is not good enough for you as a publishing venue, choose Wikiversity instead, but not both?

4) There are some features that appear unduly promotional. There is a link to a dot com home page of the inserter of the article. I dot not know how we handle or should handle this, whether prohibit such a link, etc. This is perhaps not so much a call to quasi-deletion but a call to make it less promotional.

5) I cannot determine the value of such an article. It seems to be a pseudo-article describing someone's browser extension. Can someone do a better analysis?

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:48, 8 October 2025 (UTC)

:2) Images for Wikicommons are being created, it will take a lot of time. and the text is not an exact one-to-one match
:3) I also mentioned that It was being created so that it is more accessible from mobile phone, which is not possible in RG or in Zenodo
:Let me clarify the purpose of uploading it to different platforms
:Zenodo - registration and to link DOI
:RG - Self Archiving 
:Wikiversity - Accessible by anyone from any device. LLMs may get trained on Wikiversity data or use these data for indexing 
:5) The paper is a result of a research project which involved a browser extension which was built to test the theory.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 01:34, 9 October 2025 (UTC)
:: I find the practice here of publishing non-identical but similar text (&quot;the text is not an exact one-to-one match&quot;) with almost the same title to be problematic. I cannot imagine this is a recommended practice in academic publishing. At a minimum, somewhere near the top, the page should say something like the following: &quot;This text is based on article ___ published at ___ but is not identical. The author of the differences/changes is ___.&quot; Everything else leads to an undesirable confusion. In academic publishing, the title of an article serves as key part of identification of the artifact.
:: As I said before, I seen nothing particularly academic article-like about the page except for external/superficial signs. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:30, 9 October 2025 (UTC)
:::That Article has been published under CC BY SA 4.0
:::And I am one of the author of the article. That gives me right to modify text and publish it under a similar name. However, I will add the disclaimer text that you have suggested. I hope that helps.  [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 06:07, 9 October 2025 (UTC)
:::: It may give you that right from the ''copyright'' perspective, but perhaps not from ''academic publishing integrity'' perspective. Unfortunately, I do not have any guideline handy; I am merely following my common (or not so common) sense. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:32, 9 October 2025 (UTC)
:: I would like to ask: was this article guided by someone from an academic institution, such as a university? Is it reviewed at least in some weak sense? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:39, 9 October 2025 (UTC)
:::Yes, This article has been reviewed by two academic professors, their names are also listed as co authors.
:::First, a project guide would help us with selecting a topic and with the document 
:::Second, an Internal examiner would go through our experiment and approve it
:::Finally, External Examiner would examine the documentation and verify it. 
:::We were required by these professors to put their name under contributions [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 05:48, 9 October 2025 (UTC)
:: Let me explicate the promotional potential of such a page a bit: one can go to the page of the article in Wikiversity --&gt; https://tomjoejames.com/ --&gt; HitMyTarget (a commercial, profit-making entity?) Why would the link be to a commercial web site rather than an academic page, or perhaps a LinkedIn account, which I think the person has? There could also be no link at all; a search for the name would turn out something in Google as well. But providing a direct link would drive users/viewers toward that website much stronger since otherwise the viewer of the page would have to open a new Google search window or the like. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:45, 9 October 2025 (UTC)
:::It is evident that the website is not even close to being complete. 
:::I will be creating a separate page under the same domain name specifically for people to contact me.
:::The url would probably be defined as tomjoejames.com/contact-me/
:::I haven't decided yet. But that is my personal website.
:::If the community requires me to remove it, I will. But personally I think people who are from here most likely to click the link to know more about me or to contact me. Either way I think my personal website serves the purpose. 
:::As for the HitMyTarget, it can be traced from any of my links. From my research gate profile, linkedin page or even my own userpage.
:::On the article I did not add any promotional content about myself, I hyperlinked only my own name. I do not know how that is promotional. [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 06:04, 9 October 2025 (UTC)
:::: I am pausing any further responses from me to see whether anyone else has any input. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:30, 9 October 2025 (UTC)
:What does it mean &quot;There is perhaps no more appearance/suspicion of copyright violation&quot;? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:57, 16 October 2025 (UTC)
:I have accepted VRT permission per [[ticket:2025100410001149]] FYI. [[User:Matrix|Matrix]] ([[User talk:Matrix|discuss]] • [[Special:Contributions/Matrix|contribs]]) 11:00, 28 October 2025 (UTC)
::Thank you Matrix [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 12:43, 28 October 2025 (UTC)
:I would '''delete''' it. 1) it states its a learning resource. It could not be a learning resource as not rewieved original research. 2) It is not an ongoing research, nor the research was performed on Wikiversity - wv is not a preprint or article database. Maybe it could be moved elsewhere withn Wikimedia domain, but I dont know where. So I would delete it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:56, 20 November 2025 (UTC)
::I would '''keep it.''' Like Dan had pointed out,  we do have article-like pages in Wikiversity, and this is not just a random pseudo science article but an article that is a report of an final year project, it has been reviewed by 3 professors whose name has been mentioned at the very beginning.   [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 14:50, 21 November 2025 (UTC)
:::I think it is not good to rate pages by appearance. It can be done on other Wikimedia projects, but it cannot be done on Wikiversity, because Wikiversity does not create a static format for presenting information, but is focused on the goal and process. Unfortunately, the goal and process do not have a uniform format. While a target article on Wikipedia or an entry on Wiktionary have some standard target format, Wikiversity does not. That is why I personally rate pages according to the goals and their assessment. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:05, 22 November 2025 (UTC)

Further reading for this nomination: [[S: Wikisource:Proposed_deletions/Archives/2025#Index:Cookie_Encryption.pdf]]; EncycloPetey handled the matter. Let me quote his wisdom on Zenodo (which I lack): &quot;This is tied to a PDF on Commons that was uploaded as &quot;own work&quot; with a CC license and a doi link to Zenodo, with no indication of where this paper was published or if it was published. Zenodo is not a publisher; it is a site for storing research and sharing papers. If Zenodo is the only place this was &quot;published&quot; then it was effectively self-published. --EncycloPetey (talk) 16:14, 15 September 2025 (UTC)&quot;

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:55, 9 October 2025 (UTC)

:Can you clarify what point are you trying to state? Didn't I already state that the article is published by me? 
:I first created the article in wikisource which I thought would be the perfect place, unfortunately they do not allow self published articles that are not notable. Then I discovered Wikiversity where they allow self published articles. That is why I created the article here. 
:Unlike in wikisource, I did follow guidelines.
:Ever since you deleted the first article, I spent time reading Wikiversity guidelines and I do think that I am following it perfectly.
:I would like to get your suggestions on how should I improve the page,  10 points would be sufficient. 
:Because your goals or intentions are confusing me very much. At first you told me that the article is exactly the same as the preprint in RG and therefore there is no use to it here. And then when I continued to optimize it for Wikiversity, you went ahead and said it is problematic according to recommended academic publishing. 
:Atleast just respond to the points that I have made whether you agree or disagree. So that I clarify and proceed to discuss points that are important and relevant 
:Have you published an research article? If yes, could you send it to me so that I can see the format you have done it [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 10:45, 9 October 2025 (UTC)
:: I am giving a chance/time to other curators/custodians to look at the matter and respond to my inputs. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:14, 9 October 2025 (UTC)
:: Incidentally, above I counted 4 questions (or more), 1 request (or more?) and 1 command (or more?). That is a behavior of a commanding entity. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:24, 9 October 2025 (UTC)

I would '''delete it''''. It's more like an academic communication than a learning resource or research.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:32, 26 October 2025 (UTC)
:: In the above post, I do not see any valid rationale for deletion: we do have article-like pages, in Wikijournals and also e.g. in [[Physics/Essays/Fedosin/Stellar Stefan–Boltzmann constant]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:59, 3 November 2025 (UTC)
:::But I do, see above. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:56, 20 November 2025 (UTC)

:it is a '''student research paper''' forming part of a learning resource on web security and encryption.
:The project was conducted as part of a final-year university course and documented as a practical study on cookie encryption and it has been reviewed by three professors. However, I will be creating a sub page for the article to elaborately describe the experiment that we have conducted and the results we got.    [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 15:57, 26 October 2025 (UTC)
::And why should w host research papers? Wikiversity is not an academic Journal nor repository. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:06, 22 November 2025 (UTC)
:::I do not wish to go through this same argument once again, I've already answered to this question several times in Dan's talk page, Colloquium. you can refer them. I am not hosting the research paper here, I have already hosted the pdf in the ResearchGate, I have published a text version in the wikiversity so that it may be useful for others. Unless you can show me how that article is totally useless, I would like to '''keep''' the article in the wikiversity.    [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 10:13, 22 November 2025 (UTC)
::::And thats the point I am having. Wikiversity is not paper repository. The only way is to publish it via WikiJournal, but they want it for Wikipedia usually. Why wikiversity should be a duplication of ResearchGate, Academia or Zenodo?
::::What I can read on [[Wikiversity:What is Wikiversity?]] policy is, that Wikiversity research &quot;...includes interpreting primary sources, forming ideas, or taking observations.&quot; The article doent look to fall into this. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:43, 22 November 2025 (UTC)
:::::Well, then how come you missed the term &quot;Learning Projects&quot;? As Jtneill had pointed out, this is a legitimate learning project. And also, I do have the VRT permission to host this article on Wikiversity.  [[ticket:2025100410001149]] . besides ResearchGate is an self-archiving platform. the document version in it is not accessibly to screen readers (usually disable people use them), Translators, and also for the mobile readers. therefore I do have valid reasons to publish this article on wikiversity. 
:::::# It is a learning project, therefore according to WIkiversity Policy, It qualifies. 
:::::# I have an explicit VRT permission to host this article on Wikiversity
:::::# Versions that are published in RG, Zenodo are documents, and they are not accessible by screen readers or mobile users. Therefore it is imperative that an article version of this paper exist on here. 
:::::Therefore this article qualifies to stay here on Wikiversity.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 11:22, 22 November 2025 (UTC)

'''Keep'''. This is a legitimate student learning project that may be of use to others. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:51, 22 November 2025 (UTC)
{{archive bottom}}

== [[Pragmatics/History]] ==
{{archive top|Deleted. Other related resources have been deleted. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:24, 17 May 2026 (UTC)}}
Another KYPark page and subpages with unclear organization scheme. Contains fairly many redlinked items. See also [[User:KYPark/Literature]], perhaps a similar concept. Unlikely to be really useful for others but KYPark. '''Move to user space'''.

As an alternative, moving to [[History of Pragmatics (KYPark)]] would make sense to me: the topic is identified using a natural-language phrase (instead of the relatively unnatural slash) and the responsible editor is indicated so that the reader knows whether to look or not. And for those who oppose the brackets (which I like): [[History of Pragmatics/KYPark]]. Or also: [[KYPark/History of Pragmatics]]. But then, searches in mainspace will see that content and the question is whether that is good. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:21, 15 October 2025 (UTC)

:What about to propose the user to write some guidelines, how other can participate instead of deleting it? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:03, 16 October 2025 (UTC)
:: I plan to move the pages to userspace as I proposed. If someone wants to ask KYPark to address the problems, they should feel free. There will be plenty of time for KYPark to address the problems while the material is in user space. After the problems are addressed, the material can be moved back to mainspace. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:38, 15 November 2025 (UTC)
:So I would '''delete''' it. In the blocked user space its useless. The user cannot improve it and Wikiversity is not free hosting service for personal pages. My believe is, that there should be just a few working pages in the users spaces. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:30, 11 March 2026 (UTC)

'''Move'''. Insufficient statement of learning objective or connection to related learning resources with insufficient current activity to stay in main space. The page was originally [[History of pragmatics]] but was moved by Dave B. Therefore, I suggest moving to [[User:KYPark/History of pragmatics]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:57, 22 November 2025 (UTC)
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== [[IMHA Research Archives]] ==
I propose to '''move to userspace''', including the subpages. I struggle to understand how Wikiversity readers are supposed to benefit from the material here and in the subpages. In the log, there is e.g. '10 February 2019 Marshallsumter discuss contribs deleted page IMHA Research Archives (content was: &quot;{&lt;nowiki/&gt;{Delete|Author request}} Thanks! -&quot;)', so the page was deleted before, but not the subpages.

We could also delete all the material if we have strong enough suspicion too much of it is copyright violation. In any case, moving to user space improves the matter a little by moving the content away from Google search. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:38, 9 November 2025 (UTC)

:Looking at some sub-pages, they can be deleted e.g., because they only consist of broken links or are largely empty. I deleted a couple but haven't been through all to check. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:27, 10 November 2025 (UTC)

As an example, let me give the wikitext content of [[IMHA Research Archives/3. Scientific litterature search, storage and use]]:
&lt;pre&gt;
==[[/Medicina Maritima - the Spanish scientific maritime health journal/]]==

==[[/PubMed/]]==

==[[/Google and Google Scholar/]]==

==[[/Zotero/]]==

==[https://www.dropbox.com/sh/d91z7bcyelfvk42/AAAkIvjtBnnFMbiU9ZLOdVL9a/Andrioti_database%20sources0310.pptx?dl=0 Maritime health web portal ressources ]==
&lt;/pre&gt;
The wikilinks are red; the external link to dropbox says &quot;You don't have access&quot;. This was made in 2016. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:04, 11 November 2025 (UTC)

:I suggest delete -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:27, 12 November 2025 (UTC)
:: I think we should avoid deletion as much as possible, instead moving to user space (bar copyvio, ethics violation, etc.). This is a good general principle. It greatly improves auditability and makes it so much easier for anyone to request undeletion since they know what content they are requesting for undeletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:52, 12 November 2025 (UTC)
:::Do not recreate Wikiversity from the educational and research project to the personal blog. That will lead to the cancelation of it by WMF. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:44, 20 November 2025 (UTC)
:::: The English Wikiversity has a long tradition of moving problematic content to user space, as per evidence collected at [[User:Dan_Polansky/About Wikiversity#Moving pages to userspace]]. If Wikimedia Foundation finds this problematic, they can start a discussion in Colloquium and state their concerns. They do not need to make explicit threats at first; they can start a discussion and explain why it is problematic. They can even do it from an anonymous IP and provide a well-articulated reasoning. And anyone else can start a discussion in Colloquium to change this tradition. I do not see why we should not want to change that tradition based on well-articulated, compelling reasoning. I see no reason why Juandev should be making threats instead of them, on a per RFD basis. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:58, 21 November 2025 (UTC)
:::: If Juandev is ''sincere'' about deleting very-low-value items ''from user space'', he should perhaps demonstrate that by asking his pages like [[:cs:Uživatel:Juandev/Problémy/Kov/Repase dvířek elektroskříně]] to be deleted; otherwise, I register a ''glaring inconsistence''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:43, 21 November 2025 (UTC)
::What was the original delate page about @[[User:Jtneill|Jtneill]]? I guess that would be crucial for the decission. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:48, 20 November 2025 (UTC)
:::@[[User:Juandev|Juandev]] the couple of pages I checked and deleted were much like @[[User:Dan Polansky|Dan Polansky]] posted above i.e., headings with empty sections and/or broken links but no substantive content. But I think each sub-page needs checking. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:59, 20 November 2025 (UTC)
::::So I'm saying that the main page usually determines what the other pages are for. But if I don't know the page because it's been deleted, or why was deleted (deletion based on the founder's request is probably not the rule), it's hard to judge. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 22:16, 20 November 2025 (UTC)
:::::I've pasted the original content of the root page: [[IMHA Research Archives#Original page]] (i.e., prior to the content being removed and deletion requested) to help understand the context for the sub-pages. In 2018, Saltrabook blanked the page, indicating that the content had been moved elsewhere, and requested page deletion. Marshallsumter then deleted the main page but not the sub-pages. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:58, 21 November 2025 (UTC)
::::::I see, so if those subpages are usefull I would keept them, if not I would delete them. I dont see a point of providing free hosting to sombody, by moving many pages to their user space. The question is if we want to host (i.e. to have in the main ns) lists of links elsewhere. I have no opinion on that. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:11, 22 November 2025 (UTC)

: Let me clarify that while many of the subpages are like the example above, [[IMHA Research Archives/Scientific litterature search, storage and use/Zotero]] is different:
:: &quot;A continuous critical and evidence based learning is a core issue in clinical practice, research, teaching, publication and prevention activities. The Zotero Program is just one of many scientific literature management programs, that should be used for these purposes. Of course one can live without such a database but it helps a lot and can save a lot of time that could be used for more interesting issues. Not only that, but it helps to create better publications and knowledge. Without this program it can be very time consuming to publish a scientific article with the requested style for the references. Further in daily practice when you want to collect and cite a few references for a specific evidence in a clinical colloquium and discussion, this program is excellent. Therefore we strongly recommend that all maritime health persons learn how to use this excellent tool in their daily maritime health practice of all different types. There are good online courses for self-instruction on how to use Zotero. For example this one: Zotero fast online course But in order to increase IMHAR´s collective scientific strength in the use of EBM we would like to give training sessions in every possible opportunity, IMHA Symposia, seminars and other types of meetings. The database is useful for personal purposes but especially also for collaborative aims. At the IMHAR meeting in Paris Oct 7th 2016 we will give an introduction to the program by showing how it can be used in the daily practice and discuss strength and weaknesses compared to other similar databases.&quot;
: Even longer is e.g. [[IMHA Research Archives/Scientific litterature search, storage and use/Medicina Maritima - the Spanish scientific maritime health journal]].
: However, that does not mean these should be salvaged. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:53, 21 November 2025 (UTC)
:{{ping|Saltrabook}} I'm wondering if you can respond here to help us decide about whether to delete the IMHA Research Archives sub-pages or perhaps move them to your user space? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:58, 17 May 2026 (UTC)

== [[Palliative medicine]] ==
{{archive top|'''Kept'''. Page has been improved. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:38, 17 May 2026 (UTC)}}

Underdeveloped and has not been improved on since 2007. Author inactive. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:42, 14 December 2025 (UTC)

:Delete, per nominator [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:16, 22 January 2026 (UTC)
:Yes, I would also expect there to be more and especially that someone would write how to use it. However, it still seems to me to be a useful thing in the sense of a syllabus, so that someone who is interested in the topic knows what information to obtain in order to get a complete picture of the topic. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:55, 16 March 2026 (UTC)
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== [[Canadian Wilderness]] ==
{{archive top|Deleted per nom. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:07, 17 May 2026 (UTC)}}
This page doesn't seem to belong to wikiversity. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 09:55, 6 February 2026 (UTC)

:In principle there could be some material useful here but in practice, I don't see what this page is adding as an educational resource. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 12:54, 6 February 2026 (UTC)
:I can see this being a useful resource to a bigger project. Maybe we could move it to the &quot;[[Wikiversity:Drafts|Draft]]&quot; namespace vs. deleting it? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:28, 6 February 2026 (UTC)
::Does anyone plan to work on it? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 01:59, 8 February 2026 (UTC)
:::Next week the page has it's 17th birthday. Ever now and than someone added to it. With a lot of work it could be a nice encyclopedic article but making it educational .... Merging it may take more work than rewriting it. Move to Draft might be the best option. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 08:58, 12 February 2026 (UTC)
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== [[LQR Control for an Inverted Pendulum]] ==
{{archive top|'''Deleted''' per nom. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:36, 17 May 2026 (UTC)}}
Underdeveloped resource, has not been edited for more than a decade. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:03, 16 March 2026 (UTC)

:Looks like a test, '''delete'''. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:30, 27 March 2026 (UTC)
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== False flag &quot;authority hack&quot; user page deletion ==
{{archive top|'''Not undeleted''', the requester dropped the request. See Wikiversity:Requests for Deletion v. 2803217.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:36, 10 May 2026 (UTC)}}
'''Undeletion requested'''

Hi, Juandev marked my user page as &quot;spam&quot; and &quot;authority hack&quot;, and deleted it. 

First, I asked him for help with &quot;time limit for new users&quot;, and he replied - I should admit I dont know, what is &quot;new user limit&quot;, but if filter blocks your page because of certain external link, you may force to save anyway and sometimes it works. It should not work, when the website is blacklisted. As of now, I am not seeing you to save page in main namespace, so try to save it without external links first. 

Then he wrote me another message: Well, I have analyzed your contribution to Wikiversity and I should point out here, that this project is not a place for advertising, so there is no way of promoting your books and authority this way. - probably referring to the intro of my About me page where I present me and my work. 

Before I could explain him the difference between the neutral information and advertising and promotion, he deleted my user page. 

Here is my answer I posted to the discussion today: 

: Hi, my About Me page is just an info page with the neutral as possible presentation of my work.
:
: There is a big difference between informing and advertising. Informing is neutrally stating that something exists and requiring no action, while advertising is a special communication form with intent to cause certain action from readers. For example, click here, click there, order this, buy that.
:
: There is no such intention, form, or terms on my info page. Just neutral information. I don't hide and I am not ashamed that I am write and author, and that is a part of the usual bio, including works. I checked your user page: &quot;I graduated from the Czech University of Life Sciences in Prague and studied information science at the Faculty of Arts of Charles University.&quot; I think that if you had written a book on Life Science, you would have mentioned that as well.
:
: Most of the Info page is about my research and AIPA Method which is a valid contribution to psychology, consciousness studies, identity theory, and personality development. Actually, my paper '''AIPA Method: A Cognitive-Phenomenological Model for Identity Reconstruction and Stabilization in Pure Awareness''' is now in the peer review procedure at Journal of Consciousness Studies.
:
: Here is a part from the Wikiversity AIPA Method page in creation (waiting for the end of the time limit for new users):

: == Introduction ==
: The AIPA Method addresses a gap in contemporary personal development and consciousness science: most evidence‑based approaches (CBT, MBSR, MBCT, standard meditation) operate at the level of mental content—reframing thoughts, observing them, or reducing their impact—rather than at the level of identity structure. In contrast, AIPA targets the structural relationship between the self and the mind, aiming at durable identity reconstruction rooted in Pure Awareness rather than symptom management.
:
: The central research question of the primary AIPA preprint is whether a structured, sequentially staged method can produce permanent identity reconstruction rooted in Pure Awareness, and how such a method compares to established approaches in scope, mechanism, and outcome.
:
: == Theoretical foundations ==
: The AIPA framework is grounded in the cognitive‑phenomenological tradition (e.g., McAdams, Varela, Metzinger, Erikson), contemporary consciousness science on minimal phenomenal experience, and qualitative methods advocacy in psychology. It builds directly on:
:* Empirical work on pure awareness and Minimal Phenomenal Experience (MPE), especially Gamma &amp; Metzinger’s large‑scale study of content‑reduced awareness states.
:* Metzinger’s proposal of minimal phenomenal experience as an entry point for a minimal unifying model of consciousness.
:* Narrative identity and partial‑self models within personality and identity theory.
: Within this backdrop, AIPA proposes Pure Awareness as a distinct, operationally specified state that can become a structural ground of identity rather than a transient meditative experience.
:
: == Experiential empiricism ==
: The empirical foundation of the AIPA Method is explicitly first‑person and experiential, combining:
:* A 22‑year longitudinal autoethnographic self‑study (2003–2025) documenting partial personality episodes, protocol use, and outcomes.
:* A 13‑year prospective verification period with zero self‑reported recurrence of targeted harmful behaviors after a dated stabilization point (1 January 2006).
:* A high‑ecological‑validity “stress test” during acute bereavement, used to examine whether non‑reactive awareness remains stable under maximal provocation.
:* Two independent practitioner cases (an Amazon‑verified report and a structured questionnaire case) providing preliminary convergent signals across cognitive, emotional, behavioral, and identity dimensions.
:
: All central constructs (Pure Awareness, partial personalities, the Switch, identity stabilization) are operationalized with explicit phenomenological and behavioral criteria intended to enable replication and future third‑person measurement.
:
: I believe this is a valid contribution to Wikiversity.
:
: Best regards, Senad [[User:Senad Dizdarević|Senad Dizdarević]]


I suggest you check the deleted user page, and see for yourself if it is &quot;spam&quot; and &quot;authority hack&quot;, or a legit author's page with one paragraph short presentation, while the rest of the page is about my research project. 

Thank you for undeleting my user page, so I can use it. 

Best regards, 

Senad Dizdarević [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:26, 2 April 2026 (UTC)

:Hi Senad,
:Welcome to Wikiversity.
:It looks like you tried adding similar content to Wikipedia and ran into similar difficulties over there ([[w:User talk:Senad Dizdarević]])? Perhaps that is what has led to you Wikiversity?
:Basically, if you'd like to collaborate and help build open educational resources, feel free to contribute to Wikiversity. But if the primary motivation is to promote your autobiographical work you're probably going to run into challenges.
:Sincerely,
:James
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:11, 3 April 2026 (UTC)
::James, Hi, and thank you for your answer. 
::Yes, in 2025, I created the autobiographic page on Wikipedia, which was removed because of the links to my books on Amazon. To admin, I explained that I did not know the rules, and agreed that page is removed. Now I know that somebody else must write a Wikipedia page for you. 
::On the deleted user page on Wikiversity, there were no links to Amazon or any other form of promotion, just neutral as possible basic presentation of my writing (one sentence) and current project (the rest of the page). 
::I created Wikiversity page to present my AIPA Method project, to invite researchers to read it, give their opinion, and conduct empirical researches in their institutions. Now, it is in a theoretical phase, and needs more empirical testing. 
::Best regards, 
::Senad [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:03, 3 April 2026 (UTC)
:::It looks to me like the primary motivation for contributing to Wikiversity is to drive traffic / search engine ranking to your website?
:::* [[User:Senad Dizdarević]]
:::* [[AIPA Method]]
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:36, 4 April 2026 (UTC)
::::No, it is not. There is no link to my website, so &quot;driving traffic to my website&quot; is not possible. 
::::For your educational purposes: 
::::Copilot &quot;AI:  [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:38, 4 April 2026 (UTC)
:::::So do you still insist of undeleting your former version of your userpage if you have created the new one? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:15, 6 April 2026 (UTC)
::::::No, because in the moment of undeletition, somebody could delete it again, and so on. Thank you for not deleting my new user page, as it is made in your user page image.  [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 08:59, 6 April 2026 (UTC)

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== Undeletion request ==

It was deleted by an admin without discussion and with untrue rationale. If people take offense with the question that doesn't mean it's not a valid question and the page was good. Please undelete the Wikidebate page [https://web.archive.org/web/20250810030352/https://en.wikiversity.org/wiki/Is_it_likely_that_Earth_has_been_visited_by_aliens_millions_of_years_ago%3F Is it likely that Earth has been visited by aliens millions of years ago?]

There are lots of sources on the subject, the wikidebate is sourced very well compared to other wikidebates and wikiversity pages, and the page is educational, useful and of good quality. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 23:57, 10 April 2026 (UTC)

:Page: [[Is it likely that Earth has been visited by aliens millions of years ago?]]
:Ping: [[User:Atcovi|Atcovi]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:21, 11 April 2026 (UTC)
:There is no need for a discussion for straight garbage-level, pseudoscientific content.
:For '''Is it likely that Earth has been visited by aliens millions of years ago?''', the flaws for this page wouldn't even take someone more than a few minutes to assess:
:* Essentially, the &quot;pro&quot; arguments unproven claims being derived from irrelevant, established facts (basically: &quot;it is likely aliens have came because Earth has existed for so long [sources proving Earth's longevity]&quot;). These are not serious, scientifically-backed arguments - these are non sequiturs. It's as if I said Wikipedia has existed longer than my existence on Earth ([https://d1wqtxts1xzle7.cloudfront.net/74351725/eyJoIjogImZiODhmYzNkODU1N2UxMWExYzUyODJiYzgzZTRmZDM4OTBjODY5YWMzMjA3NDNmOWEyZTA0ZTU3ZGYwZjAyYTkiLCAidSI6ICJodHRwczovL3B1cmUuaHZhLm5sL3dz-libre.pdf?1636354596=&amp;response-content-disposition=inline%3B+filename%3DCritical_Point_of_View_A_Wikipedia_Reade.pdf&amp;Expires=1775872055&amp;Signature=GqbUZboYRvUYWi~aW40LT5eZSHrLuDL3o0-DxAH8vSvcJcGAuyByZWLF2oHTY6GlB72TqvZxpE-v9d4gvsA6myriYqO~QQQZgWxjT2JXjUWC-yiPcTF4l~lroJSi4dY0v9eKiBcU03l-aeUdrX8~UPfi0TfW0IhsmzH-VBR6X6FrzRpIqc6uM6n9YXfr5FRB3aCqqokU690af3n0Hguaub1Zgmh9qjYYqzBS0VOOHjKTTEQnDuadX3jl5CQeXYTaeCC3H0hMeVwHlratbrnuFEKC1aN0-5znCUoSzMEg21ECzGPTrSDM1W05dcK-u0ZTCeUGKAuC-2yRFL3sY46MIw__&amp;Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA#page=157 reputable source proving ''this'' fact]), therefore it's likely that my birth took place solely for the sake of me experiencing Wikipedia (0 backing). It makes no sense and no person with at least a high school-level of intelligence would take this seriously. 
:* What is worse is that the user is being misleading with their &quot;[the page is] sourced very well&quot; claim. The sources ''themselves'' don't even back up the claims. It's just used as proof for an established concept, where the user then uses this established concept to jump to an unsupported, laughable conclusion that is pulled out of thin air. It's utterly ridiculous to even consider such a page for mainspace since it clearly violates our [[Wikiversity:Verifiability]] policy. This is, once again, pseudoscientific content that has caused our website to reduce in quality over the last few years. 
:* Going source by source, we can see that:
:#[https://web.archive.org/web/20250918011642/https://timesofindia.indiatimes.com/blogs/thebigd/compress-earths-history-into-24-hrs-humans-came-at-1158-pm-yet-killed-70-of-wildlife/ ‘Compress Earth’s history into 24 hrs. Humans came at 11:58 pm, yet killed 70% of wildlife’] is literally just a blog post which doesn't even mention aliens or extraterrestrial life. It just talks about Earth's history in accordance with the 24-hour metric of time, and the author tries to use this article as a 'piece in the puzzle' of aliens &quot;possibly&quot; visiting Earth... which, once again, is unsupported and is not backed up anywhere in the article.
:#[https://web.archive.org/web/20250808053249/https://news.cornell.edu/stories/2023/11/jurassic-worlds-might-be-easier-spot-modern-earth The Cornell article does not even remotely support the idea that &quot;aliens visited Earth&quot;]. It mentions a ''chance'' of &quot;life there [a habitable exoplanet] might not be limited to microbes, but could include creatures as large and varied as the megalosauruses or microraptors that once roamed Earth.&quot;, but again, no justification to take this article as proof that &quot;aliens may have visited us!&quot;. There's no mention of aliens visiting Earth anywhere in the article. Once again this is only proving the background premise, but not the unsupported, nonsensical &quot;alien likelihood&quot; argument that the author of this garbage page is trying to push so desperately.
:#The Parker Solar Probe WP article does not even mention aliens either. It follows the same issues as the previous argument.
:And the other page this user complained about [https://en.wikiversity.org/wiki/User_talk:Atcovi#Deletion_of_educational_page_because_of_personal_opinion on my talk page] holds almost similar, maybe even more fatal mistakes, than this one. It has nothing to do with &quot;taking offense&quot;, this is just low-quality, garbage content. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:56, 11 April 2026 (UTC)
::Why do you think pro claims are required to be proven? It's possible to object to them and these are arguments, not contextualized to be statements of proven facts. And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future. Instead of insulting others' intelligence, maybe engage with the actual reasoning rather than censoring it away. And there are lots of sources, such as [https://interestingengineering.com/science/alien-civilizations-may-have-visited-earth-millions-of-years-ago-study-says Alien Civilizations May Have Visited Earth Millions of Years Ago, Study Says] etc etc. The sources are used for the arguments themselves individually. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 12:30, 11 April 2026 (UTC)
:::Because, once again, this is not a site that caters to rampant debating for the sake of &quot;we need to employ rationality and logic to solve the world's problems&quot;, we have policies that we need to fulfill. The claims made in the pro argument clearly do not meet [[Wikiversity:Verifiability]], since you cannot verify these arguments with the sources because they are not relevant. 
:::''&quot;And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future.&quot;'' The point being is that these arguments are not supported by the sources. Even the article you mention poses the idea as a hypothetical model. This is just you twisting the article to fit your unsupported narrative. I'll bring direct quotes for you to show why the linked article does not help you:
:::* ''One problem the researchers do make sure to point out is that '''they are working with only one data point: our own behaviors and capabilities for space exploration'''. “We tried to come up with a model that would involve the fewest assumptions about sociology that we could,” Carroll-Nellenback told Business Insider. '''We have no real way of knowing the motivations of an alien civilization'''.'' --&gt; proves that this is just speculation and no evidence-based arguments have been provided for the idea that aliens likely visited Earth.
:::And I'm not sure if you read my entire response, but I ''did'' engage with your &quot;actual reasoning&quot; and exposed its weaknesses and lack of adherence to Wikiversity policies. If we allowed content that was just filled with non sequiturs we would have content that fails Wikiversity's educational objectives and reduces the overall quality of this website, hence why such a harsh stance needs to be taken. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:50, 11 April 2026 (UTC)
::::Thanks for proving that the Wikimedia ecosystem is unfit to deliberate on controversial topics. The question is entirely valid and the content is far better sourced than nearly all Wikidebates and has no genuine flaws. The only possible issue with it as far as I can see is that now that Wikidebates has been paused people can't add objections if they do have sth specific to say about the topic that's not already included on that page which already had plenty of Cons and objections.
::::The page was more educational than most of Wikiversity and it was well-sourced – wikidebates was for arguments so people were invited to make arguments based on sourced things or outlined logic and the page met [[WV:V]] and most pages on Wikiversity aren't sourced as good. Doesn't look like people can see beyond their biases and personal views here but that's more evident in the marginalization and deletion of wikidebates and the low activity in that project than these selective deletions. A constructive thing to do would be to add reasoned Cons and objections not yet on the page and people had plenty of time to do that. There are and will be other sites for free constructive rational adversarial deliberation (not a big loss in that sense). [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 16:31, 22 April 2026 (UTC)
:::::Thank you for failing to address any of my arguments and going on an unrelated, nonsensical tangent that has nothing to do with the discussion. Once you start producing work that aligns with Wikiversity's content policies instead of typing up laughable, pseudoscientific garbage, then maybe your work can be accepted and not removed. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:59, 22 April 2026 (UTC)
::::::I suggest you stop ridiculing things and learn respectfully forming genuine points about the subject at hand. {{tq|the idea as a hypothetical model}} but please learn first about what arguments are and why they're not the same as a statement of objective proven fact. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 17:18, 22 April 2026 (UTC)

==Pages by Harold Foppele==
[[User:Harold Foppele]] is locally blocked indefinitely and globally banned for sockpuppetry. There were also WMF and local community concerns expressed about copyright violation and AI (over)use. As a result, I think the Wikiversity pages created by this account warrant review with regard what should be deleted, what should be retained etc.:
* [[Completing the square]]
* [[Number of independent spatial modes in a spherical volume]]
* [[Quantum]]
** [[Quantum/Andrew N. Jordan]]
* [[Quantum A Matter Of Size]]
* [[Quantum A Spooky Action at a Distance]]
* [[Quantum: A Walk Through the Universe]]
* [[Quantum Computing Algorithms in the NISQ Era]]
* [[Quantum Formulas Collection]]
* [[Quantum harmonic oscillator]]
* [[Quantum Matter Elements and Particles]]
* [[Quantum mechanics]]
** [[Quantum mechanics/Timeline]]
* [[Quantum mechanics learning module]]
* [[Quantum mechanics measurements]]
* [[Quantum Noisy Qubits]]
* [[Quantum optics beam splitter experiments]]
* [[Quantum: The Secret of Cohesion: How Waves Hold Matter Together]]
* [[Quantum Ultra fast lasers]]
* [[Speed of sound experiments]]
* [[User:Harold Foppele]]
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:12, 17 May 2026 (UTC)

:'''Delete all''' Not worth keeping. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:27, 17 May 2026 (UTC)

== [[Classical guitar pedagogy]] ==

According to the talk page, the author of this page intended to create this page for Wikipedia. At this moment in time (nearly 20 years later), the page is still riddled with red links and doesn't seem to fit Wikiversity's learning modules. Therefore, I propose that this page should be deleted. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:03, 19 May 2026 (UTC)

:'''Weak delete''' This at least has &lt;em&gt;something&lt;/em&gt; that someone could use, but agreed that it's not particularly useful and not likely to be developed. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:25, 20 May 2026 (UTC)

== [[Film writing]] ==

Undeveloped since 2007. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:05, 19 May 2026 (UTC)</text>
      <sha1>i2ge7u3jyfcgxnler4n8flhn6iee08q</sha1>
    </revision>
    <revision>
      <id>2810517</id>
      <parentid>2810516</parentid>
      <timestamp>2026-05-20T00:25:35Z</timestamp>
      <contributor>
        <username>Koavf</username>
        <id>147</id>
      </contributor>
      <comment>/* Film writing */ Reply</comment>
      <origin>2810517</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="58645" sha1="l0qbzs96r4u0jtzum40kwkguya9wp1g" xml:space="preserve">{{/header}}
== [[Korean/Words]] ==
(I go to RfD instead of ''proposed deletion'' since many pages are affected.)

I proposed to quasi-delete, i.e. '''move to userspace''' of the main (or sole?) creator, {{User|KYPark}}.

The page is organized a little bit like a dictionary. It makes it redundant to Wiktionary except that Wikiversity allows original research and there does seem to be original research there. Thus, its being organized as a dictionary would alone not necessarily be a problem.

Where I see a problem is in the organization and execution/implementation. Consider [[Korean/Words/가다]], which seems rather typical of the subpages (some subpages are like categories and transclude the pages for individual words):
* On the putative definition line, there is this: &quot;한곳에서 다른 곳으로 장소를 이동하다&quot;, apparently(?) in Korean. That does not seem to fit well into the ''English'' Wikiversity.
* There seems to be some original research into etymological relations between Korean and European languages in the &quot;Comparatives&quot; section (from what I recall, the English Wiktionary rejected this kind of content from KYPark). Admittedly, it is marked using &quot;This is a primary, secondary and/or original Eurasiatic research project at Wikiversity&quot;, so it could be tolerable, but even so, one has to wonder whether Wikiversity wants this kind of fringe science/research or outright pseudo-science.
** Fringe science: fringe physics has been moved to user space before. This would be fringe etymology. But then, original research is allowed.

Deletion is not required; moving to user space suffices, I think. Alternatively, one could at least rename the pages to make it clear from the title that this is not Wikiversity voice but rather KYPark voice, e.g. &quot;Korean/Words (KYPark)/...&quot; or  &quot;Korean/Words/KYPark/...&quot;  (recall the &quot;Fedosin&quot; pages featuring the name &quot;Fedosin&quot;).

Methodology: I see almost no methodological notes spanning the words at [[Korean/Words]]. And yet, if this is original research inventing new etymological connections, surely there should be some general considerations/analysis on how to proceed and how that manner of procedure differs from mainstream etymology?

Prefix index (max 200 items?):
{{collapse top}}
{{Small START}}
{{Special:Prefixindex/Korean/Words}}
{{Small END}}
{{Collapse bottom}}
--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:33, 24 September 2025 (UTC)

:I would keep it. If there is a course of Korean, why not to have a resesearch on Korean vocabulary? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:53, 16 October 2025 (UTC)
:: I propose to dismiss the above input: 1) it does not contain any argument, except for a question, and a question is not an argument (it can be so reinterpreted, but that includes additional burden on the interpreters, in interpreting it the wrong way); 2) it ignores all the issues I have raised, including that there is something like definition lines in Korean, in this ''English'' Wikiversity. To answer the question asked: there can be a research on Korean vocabulary in the mainspace, but not one showing the defects I identified above. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:35, 15 November 2025 (UTC)
:I've reviewed a sample of approximately 20 of the Korean/Words sub-pages and lean towards moving to user space because:
:* The pages appear to be an idiosynchratic collection of etymological pages about Korean language
:* There is minimal English instruction which is problematic for English Wikiversity
:* There is no explanation of research method
:* There is no educational rationale
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:31, 22 November 2025 (UTC)
:Well, since the original creator has indef I change my mind and I would '''delete''' it. The case is nobody knows how to continue with the research and if we move it to the userspace, the user cannot improve it eihter. What the original user can do to request admin, to send them a contentent to their email for example if they really want to improve the resource elsewhere. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:38, 11 March 2026 (UTC)

I think the consensus here is delete. {{U|Codename Noreste}} do you know an efficient way to mass delete these pages? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 17 May 2026 (UTC)
: I would use a script, but I would probably not delete those pages yet until we have the pseudobot user group. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:53, 17 May 2026 (UTC)

== [[Enhancing Web Browser Security through Cookie Encryption]] ==
{{archive top|'''Kept'''. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 17 May 2026 (UTC)}}
To avoid further conflict with the user who entered this text into Wikiversity, I am opening a RFD request.

I am not sure about how to proceed, although I am inclined to move it out of mainspace = quasi-delete. I am looking forward to get input from others, especially curators and custodians. Some considerations:

1) There is perhaps no more appearance/suspicion of copyright violation, now that the ResearchGate (RG) article (of which this is a copy, perhaps an incomplete copy?) carries a license.

2) The article is not a complete replica from RG: at a minimum, it lacks images. The inserter could have continued editing the page in his user space before he uploads images, that is, before he finalizes the page for consumption, but that did not happen. I did not check whether the text is an exact one-to-one match; the article does not indicate anything in that regard.

3) The principle implied seems to be this: users should feel free to duplicate non-peer-reviewed articles from RG in English Wikiversity, perhaps to increase the Google search and LLM yield. I find this problematic, in part for the duplication. I would say: choose a venue and publish it there. If RG is not good enough for you as a publishing venue, choose Wikiversity instead, but not both?

4) There are some features that appear unduly promotional. There is a link to a dot com home page of the inserter of the article. I dot not know how we handle or should handle this, whether prohibit such a link, etc. This is perhaps not so much a call to quasi-deletion but a call to make it less promotional.

5) I cannot determine the value of such an article. It seems to be a pseudo-article describing someone's browser extension. Can someone do a better analysis?

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:48, 8 October 2025 (UTC)

:2) Images for Wikicommons are being created, it will take a lot of time. and the text is not an exact one-to-one match
:3) I also mentioned that It was being created so that it is more accessible from mobile phone, which is not possible in RG or in Zenodo
:Let me clarify the purpose of uploading it to different platforms
:Zenodo - registration and to link DOI
:RG - Self Archiving 
:Wikiversity - Accessible by anyone from any device. LLMs may get trained on Wikiversity data or use these data for indexing 
:5) The paper is a result of a research project which involved a browser extension which was built to test the theory.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 01:34, 9 October 2025 (UTC)
:: I find the practice here of publishing non-identical but similar text (&quot;the text is not an exact one-to-one match&quot;) with almost the same title to be problematic. I cannot imagine this is a recommended practice in academic publishing. At a minimum, somewhere near the top, the page should say something like the following: &quot;This text is based on article ___ published at ___ but is not identical. The author of the differences/changes is ___.&quot; Everything else leads to an undesirable confusion. In academic publishing, the title of an article serves as key part of identification of the artifact.
:: As I said before, I seen nothing particularly academic article-like about the page except for external/superficial signs. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:30, 9 October 2025 (UTC)
:::That Article has been published under CC BY SA 4.0
:::And I am one of the author of the article. That gives me right to modify text and publish it under a similar name. However, I will add the disclaimer text that you have suggested. I hope that helps.  [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 06:07, 9 October 2025 (UTC)
:::: It may give you that right from the ''copyright'' perspective, but perhaps not from ''academic publishing integrity'' perspective. Unfortunately, I do not have any guideline handy; I am merely following my common (or not so common) sense. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:32, 9 October 2025 (UTC)
:: I would like to ask: was this article guided by someone from an academic institution, such as a university? Is it reviewed at least in some weak sense? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:39, 9 October 2025 (UTC)
:::Yes, This article has been reviewed by two academic professors, their names are also listed as co authors.
:::First, a project guide would help us with selecting a topic and with the document 
:::Second, an Internal examiner would go through our experiment and approve it
:::Finally, External Examiner would examine the documentation and verify it. 
:::We were required by these professors to put their name under contributions [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 05:48, 9 October 2025 (UTC)
:: Let me explicate the promotional potential of such a page a bit: one can go to the page of the article in Wikiversity --&gt; https://tomjoejames.com/ --&gt; HitMyTarget (a commercial, profit-making entity?) Why would the link be to a commercial web site rather than an academic page, or perhaps a LinkedIn account, which I think the person has? There could also be no link at all; a search for the name would turn out something in Google as well. But providing a direct link would drive users/viewers toward that website much stronger since otherwise the viewer of the page would have to open a new Google search window or the like. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:45, 9 October 2025 (UTC)
:::It is evident that the website is not even close to being complete. 
:::I will be creating a separate page under the same domain name specifically for people to contact me.
:::The url would probably be defined as tomjoejames.com/contact-me/
:::I haven't decided yet. But that is my personal website.
:::If the community requires me to remove it, I will. But personally I think people who are from here most likely to click the link to know more about me or to contact me. Either way I think my personal website serves the purpose. 
:::As for the HitMyTarget, it can be traced from any of my links. From my research gate profile, linkedin page or even my own userpage.
:::On the article I did not add any promotional content about myself, I hyperlinked only my own name. I do not know how that is promotional. [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 06:04, 9 October 2025 (UTC)
:::: I am pausing any further responses from me to see whether anyone else has any input. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:30, 9 October 2025 (UTC)
:What does it mean &quot;There is perhaps no more appearance/suspicion of copyright violation&quot;? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:57, 16 October 2025 (UTC)
:I have accepted VRT permission per [[ticket:2025100410001149]] FYI. [[User:Matrix|Matrix]] ([[User talk:Matrix|discuss]] • [[Special:Contributions/Matrix|contribs]]) 11:00, 28 October 2025 (UTC)
::Thank you Matrix [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 12:43, 28 October 2025 (UTC)
:I would '''delete''' it. 1) it states its a learning resource. It could not be a learning resource as not rewieved original research. 2) It is not an ongoing research, nor the research was performed on Wikiversity - wv is not a preprint or article database. Maybe it could be moved elsewhere withn Wikimedia domain, but I dont know where. So I would delete it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:56, 20 November 2025 (UTC)
::I would '''keep it.''' Like Dan had pointed out,  we do have article-like pages in Wikiversity, and this is not just a random pseudo science article but an article that is a report of an final year project, it has been reviewed by 3 professors whose name has been mentioned at the very beginning.   [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 14:50, 21 November 2025 (UTC)
:::I think it is not good to rate pages by appearance. It can be done on other Wikimedia projects, but it cannot be done on Wikiversity, because Wikiversity does not create a static format for presenting information, but is focused on the goal and process. Unfortunately, the goal and process do not have a uniform format. While a target article on Wikipedia or an entry on Wiktionary have some standard target format, Wikiversity does not. That is why I personally rate pages according to the goals and their assessment. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:05, 22 November 2025 (UTC)

Further reading for this nomination: [[S: Wikisource:Proposed_deletions/Archives/2025#Index:Cookie_Encryption.pdf]]; EncycloPetey handled the matter. Let me quote his wisdom on Zenodo (which I lack): &quot;This is tied to a PDF on Commons that was uploaded as &quot;own work&quot; with a CC license and a doi link to Zenodo, with no indication of where this paper was published or if it was published. Zenodo is not a publisher; it is a site for storing research and sharing papers. If Zenodo is the only place this was &quot;published&quot; then it was effectively self-published. --EncycloPetey (talk) 16:14, 15 September 2025 (UTC)&quot;

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:55, 9 October 2025 (UTC)

:Can you clarify what point are you trying to state? Didn't I already state that the article is published by me? 
:I first created the article in wikisource which I thought would be the perfect place, unfortunately they do not allow self published articles that are not notable. Then I discovered Wikiversity where they allow self published articles. That is why I created the article here. 
:Unlike in wikisource, I did follow guidelines.
:Ever since you deleted the first article, I spent time reading Wikiversity guidelines and I do think that I am following it perfectly.
:I would like to get your suggestions on how should I improve the page,  10 points would be sufficient. 
:Because your goals or intentions are confusing me very much. At first you told me that the article is exactly the same as the preprint in RG and therefore there is no use to it here. And then when I continued to optimize it for Wikiversity, you went ahead and said it is problematic according to recommended academic publishing. 
:Atleast just respond to the points that I have made whether you agree or disagree. So that I clarify and proceed to discuss points that are important and relevant 
:Have you published an research article? If yes, could you send it to me so that I can see the format you have done it [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 10:45, 9 October 2025 (UTC)
:: I am giving a chance/time to other curators/custodians to look at the matter and respond to my inputs. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:14, 9 October 2025 (UTC)
:: Incidentally, above I counted 4 questions (or more), 1 request (or more?) and 1 command (or more?). That is a behavior of a commanding entity. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:24, 9 October 2025 (UTC)

I would '''delete it''''. It's more like an academic communication than a learning resource or research.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:32, 26 October 2025 (UTC)
:: In the above post, I do not see any valid rationale for deletion: we do have article-like pages, in Wikijournals and also e.g. in [[Physics/Essays/Fedosin/Stellar Stefan–Boltzmann constant]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:59, 3 November 2025 (UTC)
:::But I do, see above. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:56, 20 November 2025 (UTC)

:it is a '''student research paper''' forming part of a learning resource on web security and encryption.
:The project was conducted as part of a final-year university course and documented as a practical study on cookie encryption and it has been reviewed by three professors. However, I will be creating a sub page for the article to elaborately describe the experiment that we have conducted and the results we got.    [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 15:57, 26 October 2025 (UTC)
::And why should w host research papers? Wikiversity is not an academic Journal nor repository. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:06, 22 November 2025 (UTC)
:::I do not wish to go through this same argument once again, I've already answered to this question several times in Dan's talk page, Colloquium. you can refer them. I am not hosting the research paper here, I have already hosted the pdf in the ResearchGate, I have published a text version in the wikiversity so that it may be useful for others. Unless you can show me how that article is totally useless, I would like to '''keep''' the article in the wikiversity.    [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 10:13, 22 November 2025 (UTC)
::::And thats the point I am having. Wikiversity is not paper repository. The only way is to publish it via WikiJournal, but they want it for Wikipedia usually. Why wikiversity should be a duplication of ResearchGate, Academia or Zenodo?
::::What I can read on [[Wikiversity:What is Wikiversity?]] policy is, that Wikiversity research &quot;...includes interpreting primary sources, forming ideas, or taking observations.&quot; The article doent look to fall into this. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:43, 22 November 2025 (UTC)
:::::Well, then how come you missed the term &quot;Learning Projects&quot;? As Jtneill had pointed out, this is a legitimate learning project. And also, I do have the VRT permission to host this article on Wikiversity.  [[ticket:2025100410001149]] . besides ResearchGate is an self-archiving platform. the document version in it is not accessibly to screen readers (usually disable people use them), Translators, and also for the mobile readers. therefore I do have valid reasons to publish this article on wikiversity. 
:::::# It is a learning project, therefore according to WIkiversity Policy, It qualifies. 
:::::# I have an explicit VRT permission to host this article on Wikiversity
:::::# Versions that are published in RG, Zenodo are documents, and they are not accessible by screen readers or mobile users. Therefore it is imperative that an article version of this paper exist on here. 
:::::Therefore this article qualifies to stay here on Wikiversity.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 11:22, 22 November 2025 (UTC)

'''Keep'''. This is a legitimate student learning project that may be of use to others. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:51, 22 November 2025 (UTC)
{{archive bottom}}

== [[Pragmatics/History]] ==
{{archive top|Deleted. Other related resources have been deleted. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:24, 17 May 2026 (UTC)}}
Another KYPark page and subpages with unclear organization scheme. Contains fairly many redlinked items. See also [[User:KYPark/Literature]], perhaps a similar concept. Unlikely to be really useful for others but KYPark. '''Move to user space'''.

As an alternative, moving to [[History of Pragmatics (KYPark)]] would make sense to me: the topic is identified using a natural-language phrase (instead of the relatively unnatural slash) and the responsible editor is indicated so that the reader knows whether to look or not. And for those who oppose the brackets (which I like): [[History of Pragmatics/KYPark]]. Or also: [[KYPark/History of Pragmatics]]. But then, searches in mainspace will see that content and the question is whether that is good. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:21, 15 October 2025 (UTC)

:What about to propose the user to write some guidelines, how other can participate instead of deleting it? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:03, 16 October 2025 (UTC)
:: I plan to move the pages to userspace as I proposed. If someone wants to ask KYPark to address the problems, they should feel free. There will be plenty of time for KYPark to address the problems while the material is in user space. After the problems are addressed, the material can be moved back to mainspace. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:38, 15 November 2025 (UTC)
:So I would '''delete''' it. In the blocked user space its useless. The user cannot improve it and Wikiversity is not free hosting service for personal pages. My believe is, that there should be just a few working pages in the users spaces. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:30, 11 March 2026 (UTC)

'''Move'''. Insufficient statement of learning objective or connection to related learning resources with insufficient current activity to stay in main space. The page was originally [[History of pragmatics]] but was moved by Dave B. Therefore, I suggest moving to [[User:KYPark/History of pragmatics]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:57, 22 November 2025 (UTC)
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== [[IMHA Research Archives]] ==
I propose to '''move to userspace''', including the subpages. I struggle to understand how Wikiversity readers are supposed to benefit from the material here and in the subpages. In the log, there is e.g. '10 February 2019 Marshallsumter discuss contribs deleted page IMHA Research Archives (content was: &quot;{&lt;nowiki/&gt;{Delete|Author request}} Thanks! -&quot;)', so the page was deleted before, but not the subpages.

We could also delete all the material if we have strong enough suspicion too much of it is copyright violation. In any case, moving to user space improves the matter a little by moving the content away from Google search. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:38, 9 November 2025 (UTC)

:Looking at some sub-pages, they can be deleted e.g., because they only consist of broken links or are largely empty. I deleted a couple but haven't been through all to check. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:27, 10 November 2025 (UTC)

As an example, let me give the wikitext content of [[IMHA Research Archives/3. Scientific litterature search, storage and use]]:
&lt;pre&gt;
==[[/Medicina Maritima - the Spanish scientific maritime health journal/]]==

==[[/PubMed/]]==

==[[/Google and Google Scholar/]]==

==[[/Zotero/]]==

==[https://www.dropbox.com/sh/d91z7bcyelfvk42/AAAkIvjtBnnFMbiU9ZLOdVL9a/Andrioti_database%20sources0310.pptx?dl=0 Maritime health web portal ressources ]==
&lt;/pre&gt;
The wikilinks are red; the external link to dropbox says &quot;You don't have access&quot;. This was made in 2016. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:04, 11 November 2025 (UTC)

:I suggest delete -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:27, 12 November 2025 (UTC)
:: I think we should avoid deletion as much as possible, instead moving to user space (bar copyvio, ethics violation, etc.). This is a good general principle. It greatly improves auditability and makes it so much easier for anyone to request undeletion since they know what content they are requesting for undeletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:52, 12 November 2025 (UTC)
:::Do not recreate Wikiversity from the educational and research project to the personal blog. That will lead to the cancelation of it by WMF. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:44, 20 November 2025 (UTC)
:::: The English Wikiversity has a long tradition of moving problematic content to user space, as per evidence collected at [[User:Dan_Polansky/About Wikiversity#Moving pages to userspace]]. If Wikimedia Foundation finds this problematic, they can start a discussion in Colloquium and state their concerns. They do not need to make explicit threats at first; they can start a discussion and explain why it is problematic. They can even do it from an anonymous IP and provide a well-articulated reasoning. And anyone else can start a discussion in Colloquium to change this tradition. I do not see why we should not want to change that tradition based on well-articulated, compelling reasoning. I see no reason why Juandev should be making threats instead of them, on a per RFD basis. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:58, 21 November 2025 (UTC)
:::: If Juandev is ''sincere'' about deleting very-low-value items ''from user space'', he should perhaps demonstrate that by asking his pages like [[:cs:Uživatel:Juandev/Problémy/Kov/Repase dvířek elektroskříně]] to be deleted; otherwise, I register a ''glaring inconsistence''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:43, 21 November 2025 (UTC)
::What was the original delate page about @[[User:Jtneill|Jtneill]]? I guess that would be crucial for the decission. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:48, 20 November 2025 (UTC)
:::@[[User:Juandev|Juandev]] the couple of pages I checked and deleted were much like @[[User:Dan Polansky|Dan Polansky]] posted above i.e., headings with empty sections and/or broken links but no substantive content. But I think each sub-page needs checking. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:59, 20 November 2025 (UTC)
::::So I'm saying that the main page usually determines what the other pages are for. But if I don't know the page because it's been deleted, or why was deleted (deletion based on the founder's request is probably not the rule), it's hard to judge. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 22:16, 20 November 2025 (UTC)
:::::I've pasted the original content of the root page: [[IMHA Research Archives#Original page]] (i.e., prior to the content being removed and deletion requested) to help understand the context for the sub-pages. In 2018, Saltrabook blanked the page, indicating that the content had been moved elsewhere, and requested page deletion. Marshallsumter then deleted the main page but not the sub-pages. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:58, 21 November 2025 (UTC)
::::::I see, so if those subpages are usefull I would keept them, if not I would delete them. I dont see a point of providing free hosting to sombody, by moving many pages to their user space. The question is if we want to host (i.e. to have in the main ns) lists of links elsewhere. I have no opinion on that. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:11, 22 November 2025 (UTC)

: Let me clarify that while many of the subpages are like the example above, [[IMHA Research Archives/Scientific litterature search, storage and use/Zotero]] is different:
:: &quot;A continuous critical and evidence based learning is a core issue in clinical practice, research, teaching, publication and prevention activities. The Zotero Program is just one of many scientific literature management programs, that should be used for these purposes. Of course one can live without such a database but it helps a lot and can save a lot of time that could be used for more interesting issues. Not only that, but it helps to create better publications and knowledge. Without this program it can be very time consuming to publish a scientific article with the requested style for the references. Further in daily practice when you want to collect and cite a few references for a specific evidence in a clinical colloquium and discussion, this program is excellent. Therefore we strongly recommend that all maritime health persons learn how to use this excellent tool in their daily maritime health practice of all different types. There are good online courses for self-instruction on how to use Zotero. For example this one: Zotero fast online course But in order to increase IMHAR´s collective scientific strength in the use of EBM we would like to give training sessions in every possible opportunity, IMHA Symposia, seminars and other types of meetings. The database is useful for personal purposes but especially also for collaborative aims. At the IMHAR meeting in Paris Oct 7th 2016 we will give an introduction to the program by showing how it can be used in the daily practice and discuss strength and weaknesses compared to other similar databases.&quot;
: Even longer is e.g. [[IMHA Research Archives/Scientific litterature search, storage and use/Medicina Maritima - the Spanish scientific maritime health journal]].
: However, that does not mean these should be salvaged. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:53, 21 November 2025 (UTC)
:{{ping|Saltrabook}} I'm wondering if you can respond here to help us decide about whether to delete the IMHA Research Archives sub-pages or perhaps move them to your user space? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:58, 17 May 2026 (UTC)

== [[Palliative medicine]] ==
{{archive top|'''Kept'''. Page has been improved. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:38, 17 May 2026 (UTC)}}

Underdeveloped and has not been improved on since 2007. Author inactive. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:42, 14 December 2025 (UTC)

:Delete, per nominator [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:16, 22 January 2026 (UTC)
:Yes, I would also expect there to be more and especially that someone would write how to use it. However, it still seems to me to be a useful thing in the sense of a syllabus, so that someone who is interested in the topic knows what information to obtain in order to get a complete picture of the topic. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:55, 16 March 2026 (UTC)
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== [[Canadian Wilderness]] ==
{{archive top|Deleted per nom. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:07, 17 May 2026 (UTC)}}
This page doesn't seem to belong to wikiversity. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 09:55, 6 February 2026 (UTC)

:In principle there could be some material useful here but in practice, I don't see what this page is adding as an educational resource. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 12:54, 6 February 2026 (UTC)
:I can see this being a useful resource to a bigger project. Maybe we could move it to the &quot;[[Wikiversity:Drafts|Draft]]&quot; namespace vs. deleting it? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:28, 6 February 2026 (UTC)
::Does anyone plan to work on it? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 01:59, 8 February 2026 (UTC)
:::Next week the page has it's 17th birthday. Ever now and than someone added to it. With a lot of work it could be a nice encyclopedic article but making it educational .... Merging it may take more work than rewriting it. Move to Draft might be the best option. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 08:58, 12 February 2026 (UTC)
{{archive bottom}}

== [[LQR Control for an Inverted Pendulum]] ==
{{archive top|'''Deleted''' per nom. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:36, 17 May 2026 (UTC)}}
Underdeveloped resource, has not been edited for more than a decade. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:03, 16 March 2026 (UTC)

:Looks like a test, '''delete'''. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:30, 27 March 2026 (UTC)
{{archive bottom}}

== False flag &quot;authority hack&quot; user page deletion ==
{{archive top|'''Not undeleted''', the requester dropped the request. See Wikiversity:Requests for Deletion v. 2803217.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:36, 10 May 2026 (UTC)}}
'''Undeletion requested'''

Hi, Juandev marked my user page as &quot;spam&quot; and &quot;authority hack&quot;, and deleted it. 

First, I asked him for help with &quot;time limit for new users&quot;, and he replied - I should admit I dont know, what is &quot;new user limit&quot;, but if filter blocks your page because of certain external link, you may force to save anyway and sometimes it works. It should not work, when the website is blacklisted. As of now, I am not seeing you to save page in main namespace, so try to save it without external links first. 

Then he wrote me another message: Well, I have analyzed your contribution to Wikiversity and I should point out here, that this project is not a place for advertising, so there is no way of promoting your books and authority this way. - probably referring to the intro of my About me page where I present me and my work. 

Before I could explain him the difference between the neutral information and advertising and promotion, he deleted my user page. 

Here is my answer I posted to the discussion today: 

: Hi, my About Me page is just an info page with the neutral as possible presentation of my work.
:
: There is a big difference between informing and advertising. Informing is neutrally stating that something exists and requiring no action, while advertising is a special communication form with intent to cause certain action from readers. For example, click here, click there, order this, buy that.
:
: There is no such intention, form, or terms on my info page. Just neutral information. I don't hide and I am not ashamed that I am write and author, and that is a part of the usual bio, including works. I checked your user page: &quot;I graduated from the Czech University of Life Sciences in Prague and studied information science at the Faculty of Arts of Charles University.&quot; I think that if you had written a book on Life Science, you would have mentioned that as well.
:
: Most of the Info page is about my research and AIPA Method which is a valid contribution to psychology, consciousness studies, identity theory, and personality development. Actually, my paper '''AIPA Method: A Cognitive-Phenomenological Model for Identity Reconstruction and Stabilization in Pure Awareness''' is now in the peer review procedure at Journal of Consciousness Studies.
:
: Here is a part from the Wikiversity AIPA Method page in creation (waiting for the end of the time limit for new users):

: == Introduction ==
: The AIPA Method addresses a gap in contemporary personal development and consciousness science: most evidence‑based approaches (CBT, MBSR, MBCT, standard meditation) operate at the level of mental content—reframing thoughts, observing them, or reducing their impact—rather than at the level of identity structure. In contrast, AIPA targets the structural relationship between the self and the mind, aiming at durable identity reconstruction rooted in Pure Awareness rather than symptom management.
:
: The central research question of the primary AIPA preprint is whether a structured, sequentially staged method can produce permanent identity reconstruction rooted in Pure Awareness, and how such a method compares to established approaches in scope, mechanism, and outcome.
:
: == Theoretical foundations ==
: The AIPA framework is grounded in the cognitive‑phenomenological tradition (e.g., McAdams, Varela, Metzinger, Erikson), contemporary consciousness science on minimal phenomenal experience, and qualitative methods advocacy in psychology. It builds directly on:
:* Empirical work on pure awareness and Minimal Phenomenal Experience (MPE), especially Gamma &amp; Metzinger’s large‑scale study of content‑reduced awareness states.
:* Metzinger’s proposal of minimal phenomenal experience as an entry point for a minimal unifying model of consciousness.
:* Narrative identity and partial‑self models within personality and identity theory.
: Within this backdrop, AIPA proposes Pure Awareness as a distinct, operationally specified state that can become a structural ground of identity rather than a transient meditative experience.
:
: == Experiential empiricism ==
: The empirical foundation of the AIPA Method is explicitly first‑person and experiential, combining:
:* A 22‑year longitudinal autoethnographic self‑study (2003–2025) documenting partial personality episodes, protocol use, and outcomes.
:* A 13‑year prospective verification period with zero self‑reported recurrence of targeted harmful behaviors after a dated stabilization point (1 January 2006).
:* A high‑ecological‑validity “stress test” during acute bereavement, used to examine whether non‑reactive awareness remains stable under maximal provocation.
:* Two independent practitioner cases (an Amazon‑verified report and a structured questionnaire case) providing preliminary convergent signals across cognitive, emotional, behavioral, and identity dimensions.
:
: All central constructs (Pure Awareness, partial personalities, the Switch, identity stabilization) are operationalized with explicit phenomenological and behavioral criteria intended to enable replication and future third‑person measurement.
:
: I believe this is a valid contribution to Wikiversity.
:
: Best regards, Senad [[User:Senad Dizdarević|Senad Dizdarević]]


I suggest you check the deleted user page, and see for yourself if it is &quot;spam&quot; and &quot;authority hack&quot;, or a legit author's page with one paragraph short presentation, while the rest of the page is about my research project. 

Thank you for undeleting my user page, so I can use it. 

Best regards, 

Senad Dizdarević [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:26, 2 April 2026 (UTC)

:Hi Senad,
:Welcome to Wikiversity.
:It looks like you tried adding similar content to Wikipedia and ran into similar difficulties over there ([[w:User talk:Senad Dizdarević]])? Perhaps that is what has led to you Wikiversity?
:Basically, if you'd like to collaborate and help build open educational resources, feel free to contribute to Wikiversity. But if the primary motivation is to promote your autobiographical work you're probably going to run into challenges.
:Sincerely,
:James
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:11, 3 April 2026 (UTC)
::James, Hi, and thank you for your answer. 
::Yes, in 2025, I created the autobiographic page on Wikipedia, which was removed because of the links to my books on Amazon. To admin, I explained that I did not know the rules, and agreed that page is removed. Now I know that somebody else must write a Wikipedia page for you. 
::On the deleted user page on Wikiversity, there were no links to Amazon or any other form of promotion, just neutral as possible basic presentation of my writing (one sentence) and current project (the rest of the page). 
::I created Wikiversity page to present my AIPA Method project, to invite researchers to read it, give their opinion, and conduct empirical researches in their institutions. Now, it is in a theoretical phase, and needs more empirical testing. 
::Best regards, 
::Senad [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:03, 3 April 2026 (UTC)
:::It looks to me like the primary motivation for contributing to Wikiversity is to drive traffic / search engine ranking to your website?
:::* [[User:Senad Dizdarević]]
:::* [[AIPA Method]]
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:36, 4 April 2026 (UTC)
::::No, it is not. There is no link to my website, so &quot;driving traffic to my website&quot; is not possible. 
::::For your educational purposes: 
::::Copilot &quot;AI:  [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:38, 4 April 2026 (UTC)
:::::So do you still insist of undeleting your former version of your userpage if you have created the new one? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:15, 6 April 2026 (UTC)
::::::No, because in the moment of undeletition, somebody could delete it again, and so on. Thank you for not deleting my new user page, as it is made in your user page image.  [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 08:59, 6 April 2026 (UTC)

{{archive bottom}}

== Undeletion request ==

It was deleted by an admin without discussion and with untrue rationale. If people take offense with the question that doesn't mean it's not a valid question and the page was good. Please undelete the Wikidebate page [https://web.archive.org/web/20250810030352/https://en.wikiversity.org/wiki/Is_it_likely_that_Earth_has_been_visited_by_aliens_millions_of_years_ago%3F Is it likely that Earth has been visited by aliens millions of years ago?]

There are lots of sources on the subject, the wikidebate is sourced very well compared to other wikidebates and wikiversity pages, and the page is educational, useful and of good quality. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 23:57, 10 April 2026 (UTC)

:Page: [[Is it likely that Earth has been visited by aliens millions of years ago?]]
:Ping: [[User:Atcovi|Atcovi]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:21, 11 April 2026 (UTC)
:There is no need for a discussion for straight garbage-level, pseudoscientific content.
:For '''Is it likely that Earth has been visited by aliens millions of years ago?''', the flaws for this page wouldn't even take someone more than a few minutes to assess:
:* Essentially, the &quot;pro&quot; arguments unproven claims being derived from irrelevant, established facts (basically: &quot;it is likely aliens have came because Earth has existed for so long [sources proving Earth's longevity]&quot;). These are not serious, scientifically-backed arguments - these are non sequiturs. It's as if I said Wikipedia has existed longer than my existence on Earth ([https://d1wqtxts1xzle7.cloudfront.net/74351725/eyJoIjogImZiODhmYzNkODU1N2UxMWExYzUyODJiYzgzZTRmZDM4OTBjODY5YWMzMjA3NDNmOWEyZTA0ZTU3ZGYwZjAyYTkiLCAidSI6ICJodHRwczovL3B1cmUuaHZhLm5sL3dz-libre.pdf?1636354596=&amp;response-content-disposition=inline%3B+filename%3DCritical_Point_of_View_A_Wikipedia_Reade.pdf&amp;Expires=1775872055&amp;Signature=GqbUZboYRvUYWi~aW40LT5eZSHrLuDL3o0-DxAH8vSvcJcGAuyByZWLF2oHTY6GlB72TqvZxpE-v9d4gvsA6myriYqO~QQQZgWxjT2JXjUWC-yiPcTF4l~lroJSi4dY0v9eKiBcU03l-aeUdrX8~UPfi0TfW0IhsmzH-VBR6X6FrzRpIqc6uM6n9YXfr5FRB3aCqqokU690af3n0Hguaub1Zgmh9qjYYqzBS0VOOHjKTTEQnDuadX3jl5CQeXYTaeCC3H0hMeVwHlratbrnuFEKC1aN0-5znCUoSzMEg21ECzGPTrSDM1W05dcK-u0ZTCeUGKAuC-2yRFL3sY46MIw__&amp;Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA#page=157 reputable source proving ''this'' fact]), therefore it's likely that my birth took place solely for the sake of me experiencing Wikipedia (0 backing). It makes no sense and no person with at least a high school-level of intelligence would take this seriously. 
:* What is worse is that the user is being misleading with their &quot;[the page is] sourced very well&quot; claim. The sources ''themselves'' don't even back up the claims. It's just used as proof for an established concept, where the user then uses this established concept to jump to an unsupported, laughable conclusion that is pulled out of thin air. It's utterly ridiculous to even consider such a page for mainspace since it clearly violates our [[Wikiversity:Verifiability]] policy. This is, once again, pseudoscientific content that has caused our website to reduce in quality over the last few years. 
:* Going source by source, we can see that:
:#[https://web.archive.org/web/20250918011642/https://timesofindia.indiatimes.com/blogs/thebigd/compress-earths-history-into-24-hrs-humans-came-at-1158-pm-yet-killed-70-of-wildlife/ ‘Compress Earth’s history into 24 hrs. Humans came at 11:58 pm, yet killed 70% of wildlife’] is literally just a blog post which doesn't even mention aliens or extraterrestrial life. It just talks about Earth's history in accordance with the 24-hour metric of time, and the author tries to use this article as a 'piece in the puzzle' of aliens &quot;possibly&quot; visiting Earth... which, once again, is unsupported and is not backed up anywhere in the article.
:#[https://web.archive.org/web/20250808053249/https://news.cornell.edu/stories/2023/11/jurassic-worlds-might-be-easier-spot-modern-earth The Cornell article does not even remotely support the idea that &quot;aliens visited Earth&quot;]. It mentions a ''chance'' of &quot;life there [a habitable exoplanet] might not be limited to microbes, but could include creatures as large and varied as the megalosauruses or microraptors that once roamed Earth.&quot;, but again, no justification to take this article as proof that &quot;aliens may have visited us!&quot;. There's no mention of aliens visiting Earth anywhere in the article. Once again this is only proving the background premise, but not the unsupported, nonsensical &quot;alien likelihood&quot; argument that the author of this garbage page is trying to push so desperately.
:#The Parker Solar Probe WP article does not even mention aliens either. It follows the same issues as the previous argument.
:And the other page this user complained about [https://en.wikiversity.org/wiki/User_talk:Atcovi#Deletion_of_educational_page_because_of_personal_opinion on my talk page] holds almost similar, maybe even more fatal mistakes, than this one. It has nothing to do with &quot;taking offense&quot;, this is just low-quality, garbage content. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:56, 11 April 2026 (UTC)
::Why do you think pro claims are required to be proven? It's possible to object to them and these are arguments, not contextualized to be statements of proven facts. And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future. Instead of insulting others' intelligence, maybe engage with the actual reasoning rather than censoring it away. And there are lots of sources, such as [https://interestingengineering.com/science/alien-civilizations-may-have-visited-earth-millions-of-years-ago-study-says Alien Civilizations May Have Visited Earth Millions of Years Ago, Study Says] etc etc. The sources are used for the arguments themselves individually. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 12:30, 11 April 2026 (UTC)
:::Because, once again, this is not a site that caters to rampant debating for the sake of &quot;we need to employ rationality and logic to solve the world's problems&quot;, we have policies that we need to fulfill. The claims made in the pro argument clearly do not meet [[Wikiversity:Verifiability]], since you cannot verify these arguments with the sources because they are not relevant. 
:::''&quot;And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future.&quot;'' The point being is that these arguments are not supported by the sources. Even the article you mention poses the idea as a hypothetical model. This is just you twisting the article to fit your unsupported narrative. I'll bring direct quotes for you to show why the linked article does not help you:
:::* ''One problem the researchers do make sure to point out is that '''they are working with only one data point: our own behaviors and capabilities for space exploration'''. “We tried to come up with a model that would involve the fewest assumptions about sociology that we could,” Carroll-Nellenback told Business Insider. '''We have no real way of knowing the motivations of an alien civilization'''.'' --&gt; proves that this is just speculation and no evidence-based arguments have been provided for the idea that aliens likely visited Earth.
:::And I'm not sure if you read my entire response, but I ''did'' engage with your &quot;actual reasoning&quot; and exposed its weaknesses and lack of adherence to Wikiversity policies. If we allowed content that was just filled with non sequiturs we would have content that fails Wikiversity's educational objectives and reduces the overall quality of this website, hence why such a harsh stance needs to be taken. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:50, 11 April 2026 (UTC)
::::Thanks for proving that the Wikimedia ecosystem is unfit to deliberate on controversial topics. The question is entirely valid and the content is far better sourced than nearly all Wikidebates and has no genuine flaws. The only possible issue with it as far as I can see is that now that Wikidebates has been paused people can't add objections if they do have sth specific to say about the topic that's not already included on that page which already had plenty of Cons and objections.
::::The page was more educational than most of Wikiversity and it was well-sourced – wikidebates was for arguments so people were invited to make arguments based on sourced things or outlined logic and the page met [[WV:V]] and most pages on Wikiversity aren't sourced as good. Doesn't look like people can see beyond their biases and personal views here but that's more evident in the marginalization and deletion of wikidebates and the low activity in that project than these selective deletions. A constructive thing to do would be to add reasoned Cons and objections not yet on the page and people had plenty of time to do that. There are and will be other sites for free constructive rational adversarial deliberation (not a big loss in that sense). [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 16:31, 22 April 2026 (UTC)
:::::Thank you for failing to address any of my arguments and going on an unrelated, nonsensical tangent that has nothing to do with the discussion. Once you start producing work that aligns with Wikiversity's content policies instead of typing up laughable, pseudoscientific garbage, then maybe your work can be accepted and not removed. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:59, 22 April 2026 (UTC)
::::::I suggest you stop ridiculing things and learn respectfully forming genuine points about the subject at hand. {{tq|the idea as a hypothetical model}} but please learn first about what arguments are and why they're not the same as a statement of objective proven fact. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 17:18, 22 April 2026 (UTC)

==Pages by Harold Foppele==
[[User:Harold Foppele]] is locally blocked indefinitely and globally banned for sockpuppetry. There were also WMF and local community concerns expressed about copyright violation and AI (over)use. As a result, I think the Wikiversity pages created by this account warrant review with regard what should be deleted, what should be retained etc.:
* [[Completing the square]]
* [[Number of independent spatial modes in a spherical volume]]
* [[Quantum]]
** [[Quantum/Andrew N. Jordan]]
* [[Quantum A Matter Of Size]]
* [[Quantum A Spooky Action at a Distance]]
* [[Quantum: A Walk Through the Universe]]
* [[Quantum Computing Algorithms in the NISQ Era]]
* [[Quantum Formulas Collection]]
* [[Quantum harmonic oscillator]]
* [[Quantum Matter Elements and Particles]]
* [[Quantum mechanics]]
** [[Quantum mechanics/Timeline]]
* [[Quantum mechanics learning module]]
* [[Quantum mechanics measurements]]
* [[Quantum Noisy Qubits]]
* [[Quantum optics beam splitter experiments]]
* [[Quantum: The Secret of Cohesion: How Waves Hold Matter Together]]
* [[Quantum Ultra fast lasers]]
* [[Speed of sound experiments]]
* [[User:Harold Foppele]]
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:12, 17 May 2026 (UTC)

:'''Delete all''' Not worth keeping. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:27, 17 May 2026 (UTC)

== [[Classical guitar pedagogy]] ==

According to the talk page, the author of this page intended to create this page for Wikipedia. At this moment in time (nearly 20 years later), the page is still riddled with red links and doesn't seem to fit Wikiversity's learning modules. Therefore, I propose that this page should be deleted. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:03, 19 May 2026 (UTC)

:'''Weak delete''' This at least has &lt;em&gt;something&lt;/em&gt; that someone could use, but agreed that it's not particularly useful and not likely to be developed. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:25, 20 May 2026 (UTC)

== [[Film writing]] ==

Undeveloped since 2007. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:05, 19 May 2026 (UTC)

:'''Delete''' Nothing here. Great idea in principle, tho. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:25, 20 May 2026 (UTC)</text>
      <sha1>l0qbzs96r4u0jtzum40kwkguya9wp1g</sha1>
    </revision>
    <revision>
      <id>2810534</id>
      <parentid>2810517</parentid>
      <timestamp>2026-05-20T02:57:32Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Film writing */ reply ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
      <origin>2810534</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="58907" sha1="7ztg25f582jq77e6ee5dyn3uhuctem4" xml:space="preserve">{{/header}}
== [[Korean/Words]] ==
(I go to RfD instead of ''proposed deletion'' since many pages are affected.)

I proposed to quasi-delete, i.e. '''move to userspace''' of the main (or sole?) creator, {{User|KYPark}}.

The page is organized a little bit like a dictionary. It makes it redundant to Wiktionary except that Wikiversity allows original research and there does seem to be original research there. Thus, its being organized as a dictionary would alone not necessarily be a problem.

Where I see a problem is in the organization and execution/implementation. Consider [[Korean/Words/가다]], which seems rather typical of the subpages (some subpages are like categories and transclude the pages for individual words):
* On the putative definition line, there is this: &quot;한곳에서 다른 곳으로 장소를 이동하다&quot;, apparently(?) in Korean. That does not seem to fit well into the ''English'' Wikiversity.
* There seems to be some original research into etymological relations between Korean and European languages in the &quot;Comparatives&quot; section (from what I recall, the English Wiktionary rejected this kind of content from KYPark). Admittedly, it is marked using &quot;This is a primary, secondary and/or original Eurasiatic research project at Wikiversity&quot;, so it could be tolerable, but even so, one has to wonder whether Wikiversity wants this kind of fringe science/research or outright pseudo-science.
** Fringe science: fringe physics has been moved to user space before. This would be fringe etymology. But then, original research is allowed.

Deletion is not required; moving to user space suffices, I think. Alternatively, one could at least rename the pages to make it clear from the title that this is not Wikiversity voice but rather KYPark voice, e.g. &quot;Korean/Words (KYPark)/...&quot; or  &quot;Korean/Words/KYPark/...&quot;  (recall the &quot;Fedosin&quot; pages featuring the name &quot;Fedosin&quot;).

Methodology: I see almost no methodological notes spanning the words at [[Korean/Words]]. And yet, if this is original research inventing new etymological connections, surely there should be some general considerations/analysis on how to proceed and how that manner of procedure differs from mainstream etymology?

Prefix index (max 200 items?):
{{collapse top}}
{{Small START}}
{{Special:Prefixindex/Korean/Words}}
{{Small END}}
{{Collapse bottom}}
--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:33, 24 September 2025 (UTC)

:I would keep it. If there is a course of Korean, why not to have a resesearch on Korean vocabulary? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:53, 16 October 2025 (UTC)
:: I propose to dismiss the above input: 1) it does not contain any argument, except for a question, and a question is not an argument (it can be so reinterpreted, but that includes additional burden on the interpreters, in interpreting it the wrong way); 2) it ignores all the issues I have raised, including that there is something like definition lines in Korean, in this ''English'' Wikiversity. To answer the question asked: there can be a research on Korean vocabulary in the mainspace, but not one showing the defects I identified above. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:35, 15 November 2025 (UTC)
:I've reviewed a sample of approximately 20 of the Korean/Words sub-pages and lean towards moving to user space because:
:* The pages appear to be an idiosynchratic collection of etymological pages about Korean language
:* There is minimal English instruction which is problematic for English Wikiversity
:* There is no explanation of research method
:* There is no educational rationale
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:31, 22 November 2025 (UTC)
:Well, since the original creator has indef I change my mind and I would '''delete''' it. The case is nobody knows how to continue with the research and if we move it to the userspace, the user cannot improve it eihter. What the original user can do to request admin, to send them a contentent to their email for example if they really want to improve the resource elsewhere. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:38, 11 March 2026 (UTC)

I think the consensus here is delete. {{U|Codename Noreste}} do you know an efficient way to mass delete these pages? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 17 May 2026 (UTC)
: I would use a script, but I would probably not delete those pages yet until we have the pseudobot user group. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:53, 17 May 2026 (UTC)

== [[Enhancing Web Browser Security through Cookie Encryption]] ==
{{archive top|'''Kept'''. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 17 May 2026 (UTC)}}
To avoid further conflict with the user who entered this text into Wikiversity, I am opening a RFD request.

I am not sure about how to proceed, although I am inclined to move it out of mainspace = quasi-delete. I am looking forward to get input from others, especially curators and custodians. Some considerations:

1) There is perhaps no more appearance/suspicion of copyright violation, now that the ResearchGate (RG) article (of which this is a copy, perhaps an incomplete copy?) carries a license.

2) The article is not a complete replica from RG: at a minimum, it lacks images. The inserter could have continued editing the page in his user space before he uploads images, that is, before he finalizes the page for consumption, but that did not happen. I did not check whether the text is an exact one-to-one match; the article does not indicate anything in that regard.

3) The principle implied seems to be this: users should feel free to duplicate non-peer-reviewed articles from RG in English Wikiversity, perhaps to increase the Google search and LLM yield. I find this problematic, in part for the duplication. I would say: choose a venue and publish it there. If RG is not good enough for you as a publishing venue, choose Wikiversity instead, but not both?

4) There are some features that appear unduly promotional. There is a link to a dot com home page of the inserter of the article. I dot not know how we handle or should handle this, whether prohibit such a link, etc. This is perhaps not so much a call to quasi-deletion but a call to make it less promotional.

5) I cannot determine the value of such an article. It seems to be a pseudo-article describing someone's browser extension. Can someone do a better analysis?

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:48, 8 October 2025 (UTC)

:2) Images for Wikicommons are being created, it will take a lot of time. and the text is not an exact one-to-one match
:3) I also mentioned that It was being created so that it is more accessible from mobile phone, which is not possible in RG or in Zenodo
:Let me clarify the purpose of uploading it to different platforms
:Zenodo - registration and to link DOI
:RG - Self Archiving 
:Wikiversity - Accessible by anyone from any device. LLMs may get trained on Wikiversity data or use these data for indexing 
:5) The paper is a result of a research project which involved a browser extension which was built to test the theory.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 01:34, 9 October 2025 (UTC)
:: I find the practice here of publishing non-identical but similar text (&quot;the text is not an exact one-to-one match&quot;) with almost the same title to be problematic. I cannot imagine this is a recommended practice in academic publishing. At a minimum, somewhere near the top, the page should say something like the following: &quot;This text is based on article ___ published at ___ but is not identical. The author of the differences/changes is ___.&quot; Everything else leads to an undesirable confusion. In academic publishing, the title of an article serves as key part of identification of the artifact.
:: As I said before, I seen nothing particularly academic article-like about the page except for external/superficial signs. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:30, 9 October 2025 (UTC)
:::That Article has been published under CC BY SA 4.0
:::And I am one of the author of the article. That gives me right to modify text and publish it under a similar name. However, I will add the disclaimer text that you have suggested. I hope that helps.  [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 06:07, 9 October 2025 (UTC)
:::: It may give you that right from the ''copyright'' perspective, but perhaps not from ''academic publishing integrity'' perspective. Unfortunately, I do not have any guideline handy; I am merely following my common (or not so common) sense. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:32, 9 October 2025 (UTC)
:: I would like to ask: was this article guided by someone from an academic institution, such as a university? Is it reviewed at least in some weak sense? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:39, 9 October 2025 (UTC)
:::Yes, This article has been reviewed by two academic professors, their names are also listed as co authors.
:::First, a project guide would help us with selecting a topic and with the document 
:::Second, an Internal examiner would go through our experiment and approve it
:::Finally, External Examiner would examine the documentation and verify it. 
:::We were required by these professors to put their name under contributions [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 05:48, 9 October 2025 (UTC)
:: Let me explicate the promotional potential of such a page a bit: one can go to the page of the article in Wikiversity --&gt; https://tomjoejames.com/ --&gt; HitMyTarget (a commercial, profit-making entity?) Why would the link be to a commercial web site rather than an academic page, or perhaps a LinkedIn account, which I think the person has? There could also be no link at all; a search for the name would turn out something in Google as well. But providing a direct link would drive users/viewers toward that website much stronger since otherwise the viewer of the page would have to open a new Google search window or the like. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:45, 9 October 2025 (UTC)
:::It is evident that the website is not even close to being complete. 
:::I will be creating a separate page under the same domain name specifically for people to contact me.
:::The url would probably be defined as tomjoejames.com/contact-me/
:::I haven't decided yet. But that is my personal website.
:::If the community requires me to remove it, I will. But personally I think people who are from here most likely to click the link to know more about me or to contact me. Either way I think my personal website serves the purpose. 
:::As for the HitMyTarget, it can be traced from any of my links. From my research gate profile, linkedin page or even my own userpage.
:::On the article I did not add any promotional content about myself, I hyperlinked only my own name. I do not know how that is promotional. [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 06:04, 9 October 2025 (UTC)
:::: I am pausing any further responses from me to see whether anyone else has any input. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:30, 9 October 2025 (UTC)
:What does it mean &quot;There is perhaps no more appearance/suspicion of copyright violation&quot;? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:57, 16 October 2025 (UTC)
:I have accepted VRT permission per [[ticket:2025100410001149]] FYI. [[User:Matrix|Matrix]] ([[User talk:Matrix|discuss]] • [[Special:Contributions/Matrix|contribs]]) 11:00, 28 October 2025 (UTC)
::Thank you Matrix [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 12:43, 28 October 2025 (UTC)
:I would '''delete''' it. 1) it states its a learning resource. It could not be a learning resource as not rewieved original research. 2) It is not an ongoing research, nor the research was performed on Wikiversity - wv is not a preprint or article database. Maybe it could be moved elsewhere withn Wikimedia domain, but I dont know where. So I would delete it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:56, 20 November 2025 (UTC)
::I would '''keep it.''' Like Dan had pointed out,  we do have article-like pages in Wikiversity, and this is not just a random pseudo science article but an article that is a report of an final year project, it has been reviewed by 3 professors whose name has been mentioned at the very beginning.   [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 14:50, 21 November 2025 (UTC)
:::I think it is not good to rate pages by appearance. It can be done on other Wikimedia projects, but it cannot be done on Wikiversity, because Wikiversity does not create a static format for presenting information, but is focused on the goal and process. Unfortunately, the goal and process do not have a uniform format. While a target article on Wikipedia or an entry on Wiktionary have some standard target format, Wikiversity does not. That is why I personally rate pages according to the goals and their assessment. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:05, 22 November 2025 (UTC)

Further reading for this nomination: [[S: Wikisource:Proposed_deletions/Archives/2025#Index:Cookie_Encryption.pdf]]; EncycloPetey handled the matter. Let me quote his wisdom on Zenodo (which I lack): &quot;This is tied to a PDF on Commons that was uploaded as &quot;own work&quot; with a CC license and a doi link to Zenodo, with no indication of where this paper was published or if it was published. Zenodo is not a publisher; it is a site for storing research and sharing papers. If Zenodo is the only place this was &quot;published&quot; then it was effectively self-published. --EncycloPetey (talk) 16:14, 15 September 2025 (UTC)&quot;

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:55, 9 October 2025 (UTC)

:Can you clarify what point are you trying to state? Didn't I already state that the article is published by me? 
:I first created the article in wikisource which I thought would be the perfect place, unfortunately they do not allow self published articles that are not notable. Then I discovered Wikiversity where they allow self published articles. That is why I created the article here. 
:Unlike in wikisource, I did follow guidelines.
:Ever since you deleted the first article, I spent time reading Wikiversity guidelines and I do think that I am following it perfectly.
:I would like to get your suggestions on how should I improve the page,  10 points would be sufficient. 
:Because your goals or intentions are confusing me very much. At first you told me that the article is exactly the same as the preprint in RG and therefore there is no use to it here. And then when I continued to optimize it for Wikiversity, you went ahead and said it is problematic according to recommended academic publishing. 
:Atleast just respond to the points that I have made whether you agree or disagree. So that I clarify and proceed to discuss points that are important and relevant 
:Have you published an research article? If yes, could you send it to me so that I can see the format you have done it [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 10:45, 9 October 2025 (UTC)
:: I am giving a chance/time to other curators/custodians to look at the matter and respond to my inputs. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:14, 9 October 2025 (UTC)
:: Incidentally, above I counted 4 questions (or more), 1 request (or more?) and 1 command (or more?). That is a behavior of a commanding entity. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:24, 9 October 2025 (UTC)

I would '''delete it''''. It's more like an academic communication than a learning resource or research.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:32, 26 October 2025 (UTC)
:: In the above post, I do not see any valid rationale for deletion: we do have article-like pages, in Wikijournals and also e.g. in [[Physics/Essays/Fedosin/Stellar Stefan–Boltzmann constant]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:59, 3 November 2025 (UTC)
:::But I do, see above. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:56, 20 November 2025 (UTC)

:it is a '''student research paper''' forming part of a learning resource on web security and encryption.
:The project was conducted as part of a final-year university course and documented as a practical study on cookie encryption and it has been reviewed by three professors. However, I will be creating a sub page for the article to elaborately describe the experiment that we have conducted and the results we got.    [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 15:57, 26 October 2025 (UTC)
::And why should w host research papers? Wikiversity is not an academic Journal nor repository. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:06, 22 November 2025 (UTC)
:::I do not wish to go through this same argument once again, I've already answered to this question several times in Dan's talk page, Colloquium. you can refer them. I am not hosting the research paper here, I have already hosted the pdf in the ResearchGate, I have published a text version in the wikiversity so that it may be useful for others. Unless you can show me how that article is totally useless, I would like to '''keep''' the article in the wikiversity.    [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 10:13, 22 November 2025 (UTC)
::::And thats the point I am having. Wikiversity is not paper repository. The only way is to publish it via WikiJournal, but they want it for Wikipedia usually. Why wikiversity should be a duplication of ResearchGate, Academia or Zenodo?
::::What I can read on [[Wikiversity:What is Wikiversity?]] policy is, that Wikiversity research &quot;...includes interpreting primary sources, forming ideas, or taking observations.&quot; The article doent look to fall into this. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:43, 22 November 2025 (UTC)
:::::Well, then how come you missed the term &quot;Learning Projects&quot;? As Jtneill had pointed out, this is a legitimate learning project. And also, I do have the VRT permission to host this article on Wikiversity.  [[ticket:2025100410001149]] . besides ResearchGate is an self-archiving platform. the document version in it is not accessibly to screen readers (usually disable people use them), Translators, and also for the mobile readers. therefore I do have valid reasons to publish this article on wikiversity. 
:::::# It is a learning project, therefore according to WIkiversity Policy, It qualifies. 
:::::# I have an explicit VRT permission to host this article on Wikiversity
:::::# Versions that are published in RG, Zenodo are documents, and they are not accessible by screen readers or mobile users. Therefore it is imperative that an article version of this paper exist on here. 
:::::Therefore this article qualifies to stay here on Wikiversity.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 11:22, 22 November 2025 (UTC)

'''Keep'''. This is a legitimate student learning project that may be of use to others. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:51, 22 November 2025 (UTC)
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== [[Pragmatics/History]] ==
{{archive top|Deleted. Other related resources have been deleted. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:24, 17 May 2026 (UTC)}}
Another KYPark page and subpages with unclear organization scheme. Contains fairly many redlinked items. See also [[User:KYPark/Literature]], perhaps a similar concept. Unlikely to be really useful for others but KYPark. '''Move to user space'''.

As an alternative, moving to [[History of Pragmatics (KYPark)]] would make sense to me: the topic is identified using a natural-language phrase (instead of the relatively unnatural slash) and the responsible editor is indicated so that the reader knows whether to look or not. And for those who oppose the brackets (which I like): [[History of Pragmatics/KYPark]]. Or also: [[KYPark/History of Pragmatics]]. But then, searches in mainspace will see that content and the question is whether that is good. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:21, 15 October 2025 (UTC)

:What about to propose the user to write some guidelines, how other can participate instead of deleting it? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:03, 16 October 2025 (UTC)
:: I plan to move the pages to userspace as I proposed. If someone wants to ask KYPark to address the problems, they should feel free. There will be plenty of time for KYPark to address the problems while the material is in user space. After the problems are addressed, the material can be moved back to mainspace. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:38, 15 November 2025 (UTC)
:So I would '''delete''' it. In the blocked user space its useless. The user cannot improve it and Wikiversity is not free hosting service for personal pages. My believe is, that there should be just a few working pages in the users spaces. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:30, 11 March 2026 (UTC)

'''Move'''. Insufficient statement of learning objective or connection to related learning resources with insufficient current activity to stay in main space. The page was originally [[History of pragmatics]] but was moved by Dave B. Therefore, I suggest moving to [[User:KYPark/History of pragmatics]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:57, 22 November 2025 (UTC)
{{archive bottom}}

== [[IMHA Research Archives]] ==
I propose to '''move to userspace''', including the subpages. I struggle to understand how Wikiversity readers are supposed to benefit from the material here and in the subpages. In the log, there is e.g. '10 February 2019 Marshallsumter discuss contribs deleted page IMHA Research Archives (content was: &quot;{&lt;nowiki/&gt;{Delete|Author request}} Thanks! -&quot;)', so the page was deleted before, but not the subpages.

We could also delete all the material if we have strong enough suspicion too much of it is copyright violation. In any case, moving to user space improves the matter a little by moving the content away from Google search. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:38, 9 November 2025 (UTC)

:Looking at some sub-pages, they can be deleted e.g., because they only consist of broken links or are largely empty. I deleted a couple but haven't been through all to check. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:27, 10 November 2025 (UTC)

As an example, let me give the wikitext content of [[IMHA Research Archives/3. Scientific litterature search, storage and use]]:
&lt;pre&gt;
==[[/Medicina Maritima - the Spanish scientific maritime health journal/]]==

==[[/PubMed/]]==

==[[/Google and Google Scholar/]]==

==[[/Zotero/]]==

==[https://www.dropbox.com/sh/d91z7bcyelfvk42/AAAkIvjtBnnFMbiU9ZLOdVL9a/Andrioti_database%20sources0310.pptx?dl=0 Maritime health web portal ressources ]==
&lt;/pre&gt;
The wikilinks are red; the external link to dropbox says &quot;You don't have access&quot;. This was made in 2016. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:04, 11 November 2025 (UTC)

:I suggest delete -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:27, 12 November 2025 (UTC)
:: I think we should avoid deletion as much as possible, instead moving to user space (bar copyvio, ethics violation, etc.). This is a good general principle. It greatly improves auditability and makes it so much easier for anyone to request undeletion since they know what content they are requesting for undeletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:52, 12 November 2025 (UTC)
:::Do not recreate Wikiversity from the educational and research project to the personal blog. That will lead to the cancelation of it by WMF. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:44, 20 November 2025 (UTC)
:::: The English Wikiversity has a long tradition of moving problematic content to user space, as per evidence collected at [[User:Dan_Polansky/About Wikiversity#Moving pages to userspace]]. If Wikimedia Foundation finds this problematic, they can start a discussion in Colloquium and state their concerns. They do not need to make explicit threats at first; they can start a discussion and explain why it is problematic. They can even do it from an anonymous IP and provide a well-articulated reasoning. And anyone else can start a discussion in Colloquium to change this tradition. I do not see why we should not want to change that tradition based on well-articulated, compelling reasoning. I see no reason why Juandev should be making threats instead of them, on a per RFD basis. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:58, 21 November 2025 (UTC)
:::: If Juandev is ''sincere'' about deleting very-low-value items ''from user space'', he should perhaps demonstrate that by asking his pages like [[:cs:Uživatel:Juandev/Problémy/Kov/Repase dvířek elektroskříně]] to be deleted; otherwise, I register a ''glaring inconsistence''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:43, 21 November 2025 (UTC)
::What was the original delate page about @[[User:Jtneill|Jtneill]]? I guess that would be crucial for the decission. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:48, 20 November 2025 (UTC)
:::@[[User:Juandev|Juandev]] the couple of pages I checked and deleted were much like @[[User:Dan Polansky|Dan Polansky]] posted above i.e., headings with empty sections and/or broken links but no substantive content. But I think each sub-page needs checking. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:59, 20 November 2025 (UTC)
::::So I'm saying that the main page usually determines what the other pages are for. But if I don't know the page because it's been deleted, or why was deleted (deletion based on the founder's request is probably not the rule), it's hard to judge. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 22:16, 20 November 2025 (UTC)
:::::I've pasted the original content of the root page: [[IMHA Research Archives#Original page]] (i.e., prior to the content being removed and deletion requested) to help understand the context for the sub-pages. In 2018, Saltrabook blanked the page, indicating that the content had been moved elsewhere, and requested page deletion. Marshallsumter then deleted the main page but not the sub-pages. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:58, 21 November 2025 (UTC)
::::::I see, so if those subpages are usefull I would keept them, if not I would delete them. I dont see a point of providing free hosting to sombody, by moving many pages to their user space. The question is if we want to host (i.e. to have in the main ns) lists of links elsewhere. I have no opinion on that. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:11, 22 November 2025 (UTC)

: Let me clarify that while many of the subpages are like the example above, [[IMHA Research Archives/Scientific litterature search, storage and use/Zotero]] is different:
:: &quot;A continuous critical and evidence based learning is a core issue in clinical practice, research, teaching, publication and prevention activities. The Zotero Program is just one of many scientific literature management programs, that should be used for these purposes. Of course one can live without such a database but it helps a lot and can save a lot of time that could be used for more interesting issues. Not only that, but it helps to create better publications and knowledge. Without this program it can be very time consuming to publish a scientific article with the requested style for the references. Further in daily practice when you want to collect and cite a few references for a specific evidence in a clinical colloquium and discussion, this program is excellent. Therefore we strongly recommend that all maritime health persons learn how to use this excellent tool in their daily maritime health practice of all different types. There are good online courses for self-instruction on how to use Zotero. For example this one: Zotero fast online course But in order to increase IMHAR´s collective scientific strength in the use of EBM we would like to give training sessions in every possible opportunity, IMHA Symposia, seminars and other types of meetings. The database is useful for personal purposes but especially also for collaborative aims. At the IMHAR meeting in Paris Oct 7th 2016 we will give an introduction to the program by showing how it can be used in the daily practice and discuss strength and weaknesses compared to other similar databases.&quot;
: Even longer is e.g. [[IMHA Research Archives/Scientific litterature search, storage and use/Medicina Maritima - the Spanish scientific maritime health journal]].
: However, that does not mean these should be salvaged. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:53, 21 November 2025 (UTC)
:{{ping|Saltrabook}} I'm wondering if you can respond here to help us decide about whether to delete the IMHA Research Archives sub-pages or perhaps move them to your user space? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:58, 17 May 2026 (UTC)

== [[Palliative medicine]] ==
{{archive top|'''Kept'''. Page has been improved. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:38, 17 May 2026 (UTC)}}

Underdeveloped and has not been improved on since 2007. Author inactive. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:42, 14 December 2025 (UTC)

:Delete, per nominator [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:16, 22 January 2026 (UTC)
:Yes, I would also expect there to be more and especially that someone would write how to use it. However, it still seems to me to be a useful thing in the sense of a syllabus, so that someone who is interested in the topic knows what information to obtain in order to get a complete picture of the topic. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:55, 16 March 2026 (UTC)
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== [[Canadian Wilderness]] ==
{{archive top|Deleted per nom. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:07, 17 May 2026 (UTC)}}
This page doesn't seem to belong to wikiversity. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 09:55, 6 February 2026 (UTC)

:In principle there could be some material useful here but in practice, I don't see what this page is adding as an educational resource. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 12:54, 6 February 2026 (UTC)
:I can see this being a useful resource to a bigger project. Maybe we could move it to the &quot;[[Wikiversity:Drafts|Draft]]&quot; namespace vs. deleting it? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:28, 6 February 2026 (UTC)
::Does anyone plan to work on it? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 01:59, 8 February 2026 (UTC)
:::Next week the page has it's 17th birthday. Ever now and than someone added to it. With a lot of work it could be a nice encyclopedic article but making it educational .... Merging it may take more work than rewriting it. Move to Draft might be the best option. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 08:58, 12 February 2026 (UTC)
{{archive bottom}}

== [[LQR Control for an Inverted Pendulum]] ==
{{archive top|'''Deleted''' per nom. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:36, 17 May 2026 (UTC)}}
Underdeveloped resource, has not been edited for more than a decade. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:03, 16 March 2026 (UTC)

:Looks like a test, '''delete'''. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:30, 27 March 2026 (UTC)
{{archive bottom}}

== False flag &quot;authority hack&quot; user page deletion ==
{{archive top|'''Not undeleted''', the requester dropped the request. See Wikiversity:Requests for Deletion v. 2803217.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:36, 10 May 2026 (UTC)}}
'''Undeletion requested'''

Hi, Juandev marked my user page as &quot;spam&quot; and &quot;authority hack&quot;, and deleted it. 

First, I asked him for help with &quot;time limit for new users&quot;, and he replied - I should admit I dont know, what is &quot;new user limit&quot;, but if filter blocks your page because of certain external link, you may force to save anyway and sometimes it works. It should not work, when the website is blacklisted. As of now, I am not seeing you to save page in main namespace, so try to save it without external links first. 

Then he wrote me another message: Well, I have analyzed your contribution to Wikiversity and I should point out here, that this project is not a place for advertising, so there is no way of promoting your books and authority this way. - probably referring to the intro of my About me page where I present me and my work. 

Before I could explain him the difference between the neutral information and advertising and promotion, he deleted my user page. 

Here is my answer I posted to the discussion today: 

: Hi, my About Me page is just an info page with the neutral as possible presentation of my work.
:
: There is a big difference between informing and advertising. Informing is neutrally stating that something exists and requiring no action, while advertising is a special communication form with intent to cause certain action from readers. For example, click here, click there, order this, buy that.
:
: There is no such intention, form, or terms on my info page. Just neutral information. I don't hide and I am not ashamed that I am write and author, and that is a part of the usual bio, including works. I checked your user page: &quot;I graduated from the Czech University of Life Sciences in Prague and studied information science at the Faculty of Arts of Charles University.&quot; I think that if you had written a book on Life Science, you would have mentioned that as well.
:
: Most of the Info page is about my research and AIPA Method which is a valid contribution to psychology, consciousness studies, identity theory, and personality development. Actually, my paper '''AIPA Method: A Cognitive-Phenomenological Model for Identity Reconstruction and Stabilization in Pure Awareness''' is now in the peer review procedure at Journal of Consciousness Studies.
:
: Here is a part from the Wikiversity AIPA Method page in creation (waiting for the end of the time limit for new users):

: == Introduction ==
: The AIPA Method addresses a gap in contemporary personal development and consciousness science: most evidence‑based approaches (CBT, MBSR, MBCT, standard meditation) operate at the level of mental content—reframing thoughts, observing them, or reducing their impact—rather than at the level of identity structure. In contrast, AIPA targets the structural relationship between the self and the mind, aiming at durable identity reconstruction rooted in Pure Awareness rather than symptom management.
:
: The central research question of the primary AIPA preprint is whether a structured, sequentially staged method can produce permanent identity reconstruction rooted in Pure Awareness, and how such a method compares to established approaches in scope, mechanism, and outcome.
:
: == Theoretical foundations ==
: The AIPA framework is grounded in the cognitive‑phenomenological tradition (e.g., McAdams, Varela, Metzinger, Erikson), contemporary consciousness science on minimal phenomenal experience, and qualitative methods advocacy in psychology. It builds directly on:
:* Empirical work on pure awareness and Minimal Phenomenal Experience (MPE), especially Gamma &amp; Metzinger’s large‑scale study of content‑reduced awareness states.
:* Metzinger’s proposal of minimal phenomenal experience as an entry point for a minimal unifying model of consciousness.
:* Narrative identity and partial‑self models within personality and identity theory.
: Within this backdrop, AIPA proposes Pure Awareness as a distinct, operationally specified state that can become a structural ground of identity rather than a transient meditative experience.
:
: == Experiential empiricism ==
: The empirical foundation of the AIPA Method is explicitly first‑person and experiential, combining:
:* A 22‑year longitudinal autoethnographic self‑study (2003–2025) documenting partial personality episodes, protocol use, and outcomes.
:* A 13‑year prospective verification period with zero self‑reported recurrence of targeted harmful behaviors after a dated stabilization point (1 January 2006).
:* A high‑ecological‑validity “stress test” during acute bereavement, used to examine whether non‑reactive awareness remains stable under maximal provocation.
:* Two independent practitioner cases (an Amazon‑verified report and a structured questionnaire case) providing preliminary convergent signals across cognitive, emotional, behavioral, and identity dimensions.
:
: All central constructs (Pure Awareness, partial personalities, the Switch, identity stabilization) are operationalized with explicit phenomenological and behavioral criteria intended to enable replication and future third‑person measurement.
:
: I believe this is a valid contribution to Wikiversity.
:
: Best regards, Senad [[User:Senad Dizdarević|Senad Dizdarević]]


I suggest you check the deleted user page, and see for yourself if it is &quot;spam&quot; and &quot;authority hack&quot;, or a legit author's page with one paragraph short presentation, while the rest of the page is about my research project. 

Thank you for undeleting my user page, so I can use it. 

Best regards, 

Senad Dizdarević [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:26, 2 April 2026 (UTC)

:Hi Senad,
:Welcome to Wikiversity.
:It looks like you tried adding similar content to Wikipedia and ran into similar difficulties over there ([[w:User talk:Senad Dizdarević]])? Perhaps that is what has led to you Wikiversity?
:Basically, if you'd like to collaborate and help build open educational resources, feel free to contribute to Wikiversity. But if the primary motivation is to promote your autobiographical work you're probably going to run into challenges.
:Sincerely,
:James
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:11, 3 April 2026 (UTC)
::James, Hi, and thank you for your answer. 
::Yes, in 2025, I created the autobiographic page on Wikipedia, which was removed because of the links to my books on Amazon. To admin, I explained that I did not know the rules, and agreed that page is removed. Now I know that somebody else must write a Wikipedia page for you. 
::On the deleted user page on Wikiversity, there were no links to Amazon or any other form of promotion, just neutral as possible basic presentation of my writing (one sentence) and current project (the rest of the page). 
::I created Wikiversity page to present my AIPA Method project, to invite researchers to read it, give their opinion, and conduct empirical researches in their institutions. Now, it is in a theoretical phase, and needs more empirical testing. 
::Best regards, 
::Senad [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:03, 3 April 2026 (UTC)
:::It looks to me like the primary motivation for contributing to Wikiversity is to drive traffic / search engine ranking to your website?
:::* [[User:Senad Dizdarević]]
:::* [[AIPA Method]]
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:36, 4 April 2026 (UTC)
::::No, it is not. There is no link to my website, so &quot;driving traffic to my website&quot; is not possible. 
::::For your educational purposes: 
::::Copilot &quot;AI:  [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:38, 4 April 2026 (UTC)
:::::So do you still insist of undeleting your former version of your userpage if you have created the new one? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:15, 6 April 2026 (UTC)
::::::No, because in the moment of undeletition, somebody could delete it again, and so on. Thank you for not deleting my new user page, as it is made in your user page image.  [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 08:59, 6 April 2026 (UTC)

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== Undeletion request ==

It was deleted by an admin without discussion and with untrue rationale. If people take offense with the question that doesn't mean it's not a valid question and the page was good. Please undelete the Wikidebate page [https://web.archive.org/web/20250810030352/https://en.wikiversity.org/wiki/Is_it_likely_that_Earth_has_been_visited_by_aliens_millions_of_years_ago%3F Is it likely that Earth has been visited by aliens millions of years ago?]

There are lots of sources on the subject, the wikidebate is sourced very well compared to other wikidebates and wikiversity pages, and the page is educational, useful and of good quality. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 23:57, 10 April 2026 (UTC)

:Page: [[Is it likely that Earth has been visited by aliens millions of years ago?]]
:Ping: [[User:Atcovi|Atcovi]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:21, 11 April 2026 (UTC)
:There is no need for a discussion for straight garbage-level, pseudoscientific content.
:For '''Is it likely that Earth has been visited by aliens millions of years ago?''', the flaws for this page wouldn't even take someone more than a few minutes to assess:
:* Essentially, the &quot;pro&quot; arguments unproven claims being derived from irrelevant, established facts (basically: &quot;it is likely aliens have came because Earth has existed for so long [sources proving Earth's longevity]&quot;). These are not serious, scientifically-backed arguments - these are non sequiturs. It's as if I said Wikipedia has existed longer than my existence on Earth ([https://d1wqtxts1xzle7.cloudfront.net/74351725/eyJoIjogImZiODhmYzNkODU1N2UxMWExYzUyODJiYzgzZTRmZDM4OTBjODY5YWMzMjA3NDNmOWEyZTA0ZTU3ZGYwZjAyYTkiLCAidSI6ICJodHRwczovL3B1cmUuaHZhLm5sL3dz-libre.pdf?1636354596=&amp;response-content-disposition=inline%3B+filename%3DCritical_Point_of_View_A_Wikipedia_Reade.pdf&amp;Expires=1775872055&amp;Signature=GqbUZboYRvUYWi~aW40LT5eZSHrLuDL3o0-DxAH8vSvcJcGAuyByZWLF2oHTY6GlB72TqvZxpE-v9d4gvsA6myriYqO~QQQZgWxjT2JXjUWC-yiPcTF4l~lroJSi4dY0v9eKiBcU03l-aeUdrX8~UPfi0TfW0IhsmzH-VBR6X6FrzRpIqc6uM6n9YXfr5FRB3aCqqokU690af3n0Hguaub1Zgmh9qjYYqzBS0VOOHjKTTEQnDuadX3jl5CQeXYTaeCC3H0hMeVwHlratbrnuFEKC1aN0-5znCUoSzMEg21ECzGPTrSDM1W05dcK-u0ZTCeUGKAuC-2yRFL3sY46MIw__&amp;Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA#page=157 reputable source proving ''this'' fact]), therefore it's likely that my birth took place solely for the sake of me experiencing Wikipedia (0 backing). It makes no sense and no person with at least a high school-level of intelligence would take this seriously. 
:* What is worse is that the user is being misleading with their &quot;[the page is] sourced very well&quot; claim. The sources ''themselves'' don't even back up the claims. It's just used as proof for an established concept, where the user then uses this established concept to jump to an unsupported, laughable conclusion that is pulled out of thin air. It's utterly ridiculous to even consider such a page for mainspace since it clearly violates our [[Wikiversity:Verifiability]] policy. This is, once again, pseudoscientific content that has caused our website to reduce in quality over the last few years. 
:* Going source by source, we can see that:
:#[https://web.archive.org/web/20250918011642/https://timesofindia.indiatimes.com/blogs/thebigd/compress-earths-history-into-24-hrs-humans-came-at-1158-pm-yet-killed-70-of-wildlife/ ‘Compress Earth’s history into 24 hrs. Humans came at 11:58 pm, yet killed 70% of wildlife’] is literally just a blog post which doesn't even mention aliens or extraterrestrial life. It just talks about Earth's history in accordance with the 24-hour metric of time, and the author tries to use this article as a 'piece in the puzzle' of aliens &quot;possibly&quot; visiting Earth... which, once again, is unsupported and is not backed up anywhere in the article.
:#[https://web.archive.org/web/20250808053249/https://news.cornell.edu/stories/2023/11/jurassic-worlds-might-be-easier-spot-modern-earth The Cornell article does not even remotely support the idea that &quot;aliens visited Earth&quot;]. It mentions a ''chance'' of &quot;life there [a habitable exoplanet] might not be limited to microbes, but could include creatures as large and varied as the megalosauruses or microraptors that once roamed Earth.&quot;, but again, no justification to take this article as proof that &quot;aliens may have visited us!&quot;. There's no mention of aliens visiting Earth anywhere in the article. Once again this is only proving the background premise, but not the unsupported, nonsensical &quot;alien likelihood&quot; argument that the author of this garbage page is trying to push so desperately.
:#The Parker Solar Probe WP article does not even mention aliens either. It follows the same issues as the previous argument.
:And the other page this user complained about [https://en.wikiversity.org/wiki/User_talk:Atcovi#Deletion_of_educational_page_because_of_personal_opinion on my talk page] holds almost similar, maybe even more fatal mistakes, than this one. It has nothing to do with &quot;taking offense&quot;, this is just low-quality, garbage content. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:56, 11 April 2026 (UTC)
::Why do you think pro claims are required to be proven? It's possible to object to them and these are arguments, not contextualized to be statements of proven facts. And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future. Instead of insulting others' intelligence, maybe engage with the actual reasoning rather than censoring it away. And there are lots of sources, such as [https://interestingengineering.com/science/alien-civilizations-may-have-visited-earth-millions-of-years-ago-study-says Alien Civilizations May Have Visited Earth Millions of Years Ago, Study Says] etc etc. The sources are used for the arguments themselves individually. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 12:30, 11 April 2026 (UTC)
:::Because, once again, this is not a site that caters to rampant debating for the sake of &quot;we need to employ rationality and logic to solve the world's problems&quot;, we have policies that we need to fulfill. The claims made in the pro argument clearly do not meet [[Wikiversity:Verifiability]], since you cannot verify these arguments with the sources because they are not relevant. 
:::''&quot;And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future.&quot;'' The point being is that these arguments are not supported by the sources. Even the article you mention poses the idea as a hypothetical model. This is just you twisting the article to fit your unsupported narrative. I'll bring direct quotes for you to show why the linked article does not help you:
:::* ''One problem the researchers do make sure to point out is that '''they are working with only one data point: our own behaviors and capabilities for space exploration'''. “We tried to come up with a model that would involve the fewest assumptions about sociology that we could,” Carroll-Nellenback told Business Insider. '''We have no real way of knowing the motivations of an alien civilization'''.'' --&gt; proves that this is just speculation and no evidence-based arguments have been provided for the idea that aliens likely visited Earth.
:::And I'm not sure if you read my entire response, but I ''did'' engage with your &quot;actual reasoning&quot; and exposed its weaknesses and lack of adherence to Wikiversity policies. If we allowed content that was just filled with non sequiturs we would have content that fails Wikiversity's educational objectives and reduces the overall quality of this website, hence why such a harsh stance needs to be taken. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:50, 11 April 2026 (UTC)
::::Thanks for proving that the Wikimedia ecosystem is unfit to deliberate on controversial topics. The question is entirely valid and the content is far better sourced than nearly all Wikidebates and has no genuine flaws. The only possible issue with it as far as I can see is that now that Wikidebates has been paused people can't add objections if they do have sth specific to say about the topic that's not already included on that page which already had plenty of Cons and objections.
::::The page was more educational than most of Wikiversity and it was well-sourced – wikidebates was for arguments so people were invited to make arguments based on sourced things or outlined logic and the page met [[WV:V]] and most pages on Wikiversity aren't sourced as good. Doesn't look like people can see beyond their biases and personal views here but that's more evident in the marginalization and deletion of wikidebates and the low activity in that project than these selective deletions. A constructive thing to do would be to add reasoned Cons and objections not yet on the page and people had plenty of time to do that. There are and will be other sites for free constructive rational adversarial deliberation (not a big loss in that sense). [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 16:31, 22 April 2026 (UTC)
:::::Thank you for failing to address any of my arguments and going on an unrelated, nonsensical tangent that has nothing to do with the discussion. Once you start producing work that aligns with Wikiversity's content policies instead of typing up laughable, pseudoscientific garbage, then maybe your work can be accepted and not removed. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:59, 22 April 2026 (UTC)
::::::I suggest you stop ridiculing things and learn respectfully forming genuine points about the subject at hand. {{tq|the idea as a hypothetical model}} but please learn first about what arguments are and why they're not the same as a statement of objective proven fact. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 17:18, 22 April 2026 (UTC)

==Pages by Harold Foppele==
[[User:Harold Foppele]] is locally blocked indefinitely and globally banned for sockpuppetry. There were also WMF and local community concerns expressed about copyright violation and AI (over)use. As a result, I think the Wikiversity pages created by this account warrant review with regard what should be deleted, what should be retained etc.:
* [[Completing the square]]
* [[Number of independent spatial modes in a spherical volume]]
* [[Quantum]]
** [[Quantum/Andrew N. Jordan]]
* [[Quantum A Matter Of Size]]
* [[Quantum A Spooky Action at a Distance]]
* [[Quantum: A Walk Through the Universe]]
* [[Quantum Computing Algorithms in the NISQ Era]]
* [[Quantum Formulas Collection]]
* [[Quantum harmonic oscillator]]
* [[Quantum Matter Elements and Particles]]
* [[Quantum mechanics]]
** [[Quantum mechanics/Timeline]]
* [[Quantum mechanics learning module]]
* [[Quantum mechanics measurements]]
* [[Quantum Noisy Qubits]]
* [[Quantum optics beam splitter experiments]]
* [[Quantum: The Secret of Cohesion: How Waves Hold Matter Together]]
* [[Quantum Ultra fast lasers]]
* [[Speed of sound experiments]]
* [[User:Harold Foppele]]
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:12, 17 May 2026 (UTC)

:'''Delete all''' Not worth keeping. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:27, 17 May 2026 (UTC)

== [[Classical guitar pedagogy]] ==

According to the talk page, the author of this page intended to create this page for Wikipedia. At this moment in time (nearly 20 years later), the page is still riddled with red links and doesn't seem to fit Wikiversity's learning modules. Therefore, I propose that this page should be deleted. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:03, 19 May 2026 (UTC)

:'''Weak delete''' This at least has &lt;em&gt;something&lt;/em&gt; that someone could use, but agreed that it's not particularly useful and not likely to be developed. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:25, 20 May 2026 (UTC)

== [[Film writing]] ==

Undeveloped since 2007. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:05, 19 May 2026 (UTC)

:'''Delete''' Nothing here. Great idea in principle, tho. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:25, 20 May 2026 (UTC)
: '''Keep''' and integrate with existing [[:Category:Filmmaking]] resources. I've tidied the page, so it looks less abandoned. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:57, 20 May 2026 (UTC)</text>
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      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
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      <comment>Replacing with a talk header.</comment>
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  <page>
    <title>Wikiversity:Bots</title>
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      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <comment>/* Currently flagged bots */ Switching to a template. (using [[wikt:MediaWiki:Gadget-AjaxEdit.js|AjaxEdit]])</comment>
      <origin>2810488</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4026" sha1="a9kfa4xm38zwc2f1o5s6inq7rb78u6a" xml:space="preserve">{{policy|WV:BOT}}
[[File:Wikiversity Bots2.svg|230px|right]]
'''Robots''' or '''bots''' are semi-automatic or automatic [[w:process (computing)|processes]] that interact with Wikiversity as though they were a human editor. They are used to manage systematic or boring tasks. Please read the guidelines below before designing and implementing any bot on Wikiversity.

== Summary ==
{{Administering Wikiversity}}
A bot does repetitive tasks in a predictable fashion, and it can be done in such a way as to be healthy for the community.

== Expectations ==
The operation of a bot requires approval. Any user can submit a request at [[Wikiversity:Bots/Status]] to obtain bot status, which prevents cluttering [[Special:Recentchanges|recent changes]]. Bot status is given by [[m:Bureaucrat|Bureaucrats]] about seven days after the request has been made, provided there has been no objection to the request by another user. The decision lies with the bureaucrats.

The bot flag ''may'' be removed after 2 or more years of bot inactivity. The bot operator will be notified on their talk page and the flag will be re-granted if the operator expresses an interest in resuming bot activity.

The burden of proof is on the bot-maker to demonstrate that the bot:
* is harmless
* is useful
* is not a server hog
* has been approved

'''Bot operators ''must'':'''
# create a separate account for bot operation.
# indicate on their bot's user page:
#* which program / language is used. ([[meta:Using the python wikipediabot|Pywikipedia]], Ruby, JavaScript...)
#* who the owner is.
# discuss with other Wikiversity editors before running their bot. Any bot request has to reach consensus to be fulfilled.
# log in with their own user account when talking to other users.

'''Bot operators ''should'':'''
* choose a name containing the word &quot;bot&quot; so that editors realize they are dealing with an [[w:automaton|automaton]].

'''Bot operators ''may'':'''
* run their bot without the bot status '''only''' during its request for approval '''and''' if asked so by another editor to check how the bot works. When testing, bot operators must delay 60 seconds between edits.

'''Bot operators are ''encouraged'' to:'''
* Release their bot's source code.
* Program their bot to stop editing if someone leaves a message on its talk page. This can be checked by looking for the &quot;You have new messages...&quot; banner in the HTML for the edit form.
* Program their bot to stop editing if they detect that they have logged out. This can be checked by looking for bot's name in the HTML for the edit form. If the bot is not logged in, then the bot's name won't be listed in the HTML. Bots running anonymously may be blocked.

== Edit rate guidelines ==
Bots running without a bot flag should edit at intervals of over 1 minute. Once they have been authorised and appropriately flagged, they should operate at an absolute minimum interval of 5 seconds (12 edits per minute). Bots should try to avoid running during the busiest hours, as they rapidly use server resources that should be reserved for human readers and editors. During these hours, they should operate at intervals of 20 seconds (3 edits per minute) to conserve resources.

Bots' editing speeds can be automatically adjusted based on server load (slave database server lag) by appending an extra parameter to the query string of each requested URL; see [[mw:Manual:Maxlag parameter]].

== Bot functionalities ==
Although a bot can be assigned a sysop flag, these requests are discouraged. The Curator flag may be assigned to a bot operated by a Custodian.

==Currently flagged bots==
{{List of bots}}

==Request help from a bot==
Do you have a task that would benefit from a bot's help? Post a request at [[Wikiversity:Colloquium]].

==Requests for bot status==
Are you interested in helping Wikiversity by operating a bot? Please see [[Wikiversity:Bots/Status]]

==See also==
* [[meta:Bot_policy|Wikimedia Bot policy]]
* [[mw:Help:Bots|MediaWiki Bots]]

{{Official policies}}
{{Proposed policies}}</text>
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      <id>2810490</id>
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      <timestamp>2026-05-19T19:05:18Z</timestamp>
      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <comment>/* Leaderbot */ archive to [[Wikiversity:Bots/Status/Archive#Leaderbot]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
      <origin>2810490</origin>
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      <text bytes="8488" sha1="ewmileg7i2e0t9ixnfafxhi6isqxbc4" xml:space="preserve">
{{mbox
| text = Requests for the [[m:bot|bot]] flag should be made on this page. This wiki uses the [[m:bot policy|standard bot policy]], and allows [[m:bot policy#Global_bots|global bots]] and [[m:bot policy#Automatic_approval|automatic approval of certain types of bots]]. Other bots should apply below.
}}
''This page allows bot operators to ask for approval to use a bot on the English-language Wikiversity.''

'''Please read [[Wikiversity:Bots|Wikiversity policy about bots]] first.'''

Bureaucrats are able to give and remove bot status using the [[Special:UserRights]] feature.  Request are usually handled after about 7-10 days, to allow time for comments and questions from the community.  If you do not receive a timely response to your request, feel free to leave a message on the talk page of a [http://en.wikiversity.org/w/index.php?title=Special%3AListUsers&amp;username=&amp;group=bureaucrat&amp;limit=50 Bureaucrat].

Older status requests are [[/Archive/]]d.

The wikicode below is a suggestion for formating a bot request:

&lt;pre&gt;
== BotName ==
* '''Bot name''': {{User|BotName}}
* '''Bot operator''': {{User|Name}}
* '''Automatic or manually assisted''': 
* '''Purpose of the bot''': 
* '''Edit period(s)''': 
* '''Programming language(s) (and API) used''': 
* '''Other projects that are already using this bot''': 
* '''Additional information''': 
&lt;/pre&gt;

'''Requests for bot status'''

''Add new bot requests at the bottom''

== Tule-bot ==
* '''Bot name''': {{User|Tule-bot}}
* '''Bot operator''': {{User|Tule-hog}}
* '''Automatic or manually assisted''': Automatic
* '''Purpose of the bot''': lua-formatted lists of highly transcluded templates
* '''Edit period(s)''': Weekly (could be made less frequent for WV)
* '''Programming language(s) (and API) used''': Python
* '''Other projects that are already using this bot''': Modeled on [[:w:User:Ahechtbot#Task 6|Ahechtbot's Task 6]]
* '''Additional information''': Currently [[Module:Transclusion count]] uses irrelevant data periodically copied from Wikipedia to this site's [[Special:PrefixIndex/Module:Transclusion_count/data/|data pages]]. A bot is required to keep the information up to date and site-specific. Somewhat confusingly, there is already a semi-active, unregistered, Wikiversity-based [[User:Ahechtbot|Ahechtbot]] but it doesn't have a Wikiversity version of &lt;code&gt;[[:w:User:Ahechtbot/transclusioncount.py|transclusioncount.py]]&lt;/code&gt;, and appears to use the transclusion count for Wikipedia-based templates, as is seen by [[Template:Edit fully-protected]] which I have just created, but claims to be used on 7,400 pages (the Wikipedia version's count), instead of the correct [https://linkcount.toolforge.org/index.php?project=en.wikiversity.org&amp;page=Template%3AEdit+fully-protected 0 pages]. (Another example is Wikiversity's [[Module:Babel]] which is actually only used on [https://linkcount.toolforge.org/index.php?project=en.wikiversity.org&amp;page=Module%3ABabel 118 pages], not 39,000.) My Lua capabilities are minimal-to-nonexistent, but I can scrounge by some Python and SQL, so I can take a crack at the task - if someone more qualified would prefer to do it, feel free! [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:36, 1 October 2024 (UTC)

== calris25bot ==
* '''Bot name''': [[User:CalRis25/calris25bot|calris25bot]]
* '''Bot operator''': [[User:CalRis25|CalRis25]]
* '''Automatic or manually assisted''': Automatic, although I do not understand the ''exact'' meaning of &quot;manually assisted&quot;. 
* '''Purpose of the bot''': The purpose is twofold: 1. Initial upload of the pages (2200 of 3200 articles pages plus about 300 REDIRECT-pages) for the project ''[[Illustrated Companion to the Latin Dictionary]]'' (see also the [[Illustrated Companion to the Latin Dictionary/RICH-2K/Project description|project's description]]). 2. Switching the article-pages from the initial version to the production version by removing a heading. For more see ''Additional information'' and the bot's description page.
* '''Edit period(s)''': The bot will be used twice: 1. Initial upload. 2. Switching the article-pages from the initial version to the production version.
* '''Programming language(s) (and API) used''': Pywikibot
* '''Other projects that are already using this bot''': None
* '''Additional information''': For more, see the bot's description page. 1. The initial upload will be done calling Pywikiboot's ''pagefromfile''-script (one call per page-creation) using the ''-minor''-parameter (= mark the edit as &quot;minor&quot;). 2. The switch from initial version of the article pages to production-version will use a single call of Pywikibot's ''replace''-script for ''all'' article pages (= 3200) pages. Note: I can change this to smaller chunks for each call of ''replace''. Please inform me, if that is wanted. '''{{Color|Red|NOTE: The project is ready to be launched and is only awaiting the upload of the rest of the articles, for which this bot-permit is necessary.}}''' [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 15:45, 29 October 2024 (UTC)

@[[User:CalRis25|CalRis25]]: Please see [[meta:Bot policy]]. &quot;A bot must be run using a separate account from the operator, as no human editor should be granted a bot flag.&quot; It does not appear as though you have created a separate account for your bot.

I'm also inclined to suggest that, perhaps, having a separate bot account shouldn't be necessary in this case. As I understand it, this will be a short-term deployment only used to upload and quickly modify pages. There is no current planned ongoing usage.

In my experience, if you limit your bot to one update every 10 seconds (or longer) and no more than 500 edits per day, you won't cause any issues and also won't get flagged for excessive edits. More than 500 per day will trigger an internal block of some type. I don't recall whether it's 60 minutes or 24 hours. I just know I hit it a couple of times myself.

If you want to work around the limitations and use your own account, please proceed. If you want to use true bot status, please create an account for your bot and let me know on my talk page. [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:11, 4 November 2024 (UTC)

:Hello Dave, thank your for the quick answer. Actually I ''did'' create an account (''calris25bot''), and Pywikibot uses it in the user-password-file. However, if I can go on without the permit, that is fine for me. These edits-by-script are indeed only for the beginning of the project. I will take care to a) mark the edits as &quot;minor&quot;, b) put a delay of at least 10 seconds between edits, and c) limit the edits to 500 per day. Bye. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 15:46, 4 November 2024 (UTC)
:Sorry, I just checked doing a test-run with 50 page-creation-edits. The &quot;minor&quot;-option doesn't seem to work for creation of pages (which ''does'' make sense). Is it okay, if I continue as detailed (minus the minor-flag for the page-creation edits)? I'm going to wait for your replay. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 16:13, 4 November 2024 (UTC)
::@[[User:CalRis25|CalRis25]]: Something didn't work as you expected. When I go to [[User:Calris25bot]], there is no account by that name. I can't say why. I only know it's not there. And there's no way to approve bot status on a non-account.
::To confirm, look at the edit history on the pages your bot is editing. If they show your account, the bot is operating as you. If they show something else, please provide a page link so I can check the history and see what account it is using.
::I don't think you'll run into any problems as long as you limit your efforts to 10 seconds and 500 per day. At least, when I did such things a couple of years ago, those were the limits I experienced. Good luck!
::[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 16:24, 4 November 2024 (UTC)
:::Thank you, Dave. I'll continue with the edits as described without bot-status. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 16:35, 4 November 2024 (UTC).</text>
      <sha1>ewmileg7i2e0t9ixnfafxhi6isqxbc4</sha1>
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      </contributor>
      <minor />
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      <comment>/* Alternatives to deletion */ Copyedit with assistance of ChatGPT: https://chatgpt.com/share/6a0d1825-fa38-83ec-89d5-334a0c65eb49</comment>
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      <text bytes="17926" sha1="orc1mc6ptvhqur75lul58ouieclzdv4" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These templates are recommended for use by those involved with page deletions:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for large projects and difficult decisions
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts (templates and policy guidelines under construction)

'''Don't worry about selecting the &quot;correct&quot; template''', as more than one template may be appropriate. It is more important to provide assistance by flagging resources that you think shouldn't be on Wikiversity.}}

This '''deletion guideline''' explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources or specific revisions of a resource. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion]].

== Types of deletion ==
=== Speedy ===
Resources may be subject to '''speedy deletion''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources include:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|ATD|WV:ATD}}
Deletion should be a last resort. Consider [[Wikiversity:Be bold|boldly]] improving resources and/or discussing alternative solutions such as:

{| class=&quot;wikitable&quot;
! Issue
! Possible alternatives
|-
| Broken redirects
| Redirect to a more appropriate target.
|-
| Resource is in a foreign language
| Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| Outside the scope of a learning project
| Merge into another learning project or use it to start a new learning project.
|-
| Missing or incomplete licence information
| Assist the user in providing the information or supply it if possible.
|-
| Outside [[WV:scope|Wikiversity's scope]]
| Move to another Wikimedia project or external website as appropriate. Some resources, such as personal essays, may be suitable for userspace.
|-
| Insufficient or unclear educational value
| Improve the resource, request clarification, or move it to draft or userspace for further development.
|-
| Resource is too large or too small
| Split large resources into smaller resources or reorganise them into subpages. Merge small resources into broader resources or redirect them appropriately.
|}

==Deletion templates==
To remove resources from mainspace that cannot be directly used by teachers and instructors use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draftspace or userspace.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''
*&lt;code&gt;{{tl|draft archive}}&lt;/code&gt; for proposal for '' moving under Draft:Archive/ '' (proposal is under construction)

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

Note that the deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues.  

'''IMPORTANT:''' The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}
{{Shortcut|Deletion request|Dr|DR|Rfd|RfD}} 

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].
*{{spaces|2}}''See also [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]''

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>orc1mc6ptvhqur75lul58ouieclzdv4</sha1>
    </revision>
    <revision>
      <id>2810523</id>
      <parentid>2810522</parentid>
      <timestamp>2026-05-20T02:11:49Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>/* Alternatives to deletion */</comment>
      <origin>2810523</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17922" sha1="ha5494s5zeqq84j2ihi05ms1xxf9ln5" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These templates are recommended for use by those involved with page deletions:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for large projects and difficult decisions
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts (templates and policy guidelines under construction)

'''Don't worry about selecting the &quot;correct&quot; template''', as more than one template may be appropriate. It is more important to provide assistance by flagging resources that you think shouldn't be on Wikiversity.}}

This '''deletion guideline''' explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources or specific revisions of a resource. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion]].

== Types of deletion ==
=== Speedy ===
Resources may be subject to '''speedy deletion''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources include:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Consider [[Wikiversity:Be bold|boldly]] improving resources and/or discussing alternative solutions such as:

{| class=&quot;wikitable&quot;
! Issue
! Possible alternatives
|-
| Broken redirects
| Redirect to a more appropriate target.
|-
| Resource is in a foreign language
| Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| Outside the scope of a learning project
| Merge into another learning project or use it to start a new learning project.
|-
| Missing or incomplete licence information
| Assist the user in providing the information or supply it if possible.
|-
| Outside [[WV:scope|Wikiversity's scope]]
| Move to another Wikimedia project or external website as appropriate. Some resources, such as personal essays, may be suitable for userspace.
|-
| Insufficient or unclear educational value
| Improve the resource, request clarification, or move it to draft or userspace for further development.
|-
| Resource is too large or too small
| Split large resources into smaller resources or reorganise them into subpages. Merge small resources into broader resources or redirect them appropriately.
|}

==Deletion templates==
To remove resources from mainspace that cannot be directly used by teachers and instructors use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draftspace or userspace.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''
*&lt;code&gt;{{tl|draft archive}}&lt;/code&gt; for proposal for '' moving under Draft:Archive/ '' (proposal is under construction)

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

Note that the deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues.  

'''IMPORTANT:''' The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}
{{Shortcut|Deletion request|Dr|DR|Rfd|RfD}} 

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].
*{{spaces|2}}''See also [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]''

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>ha5494s5zeqq84j2ihi05ms1xxf9ln5</sha1>
    </revision>
    <revision>
      <id>2810524</id>
      <parentid>2810523</parentid>
      <timestamp>2026-05-20T02:18:00Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Alternatives to deletion */ Reorder table; reintroduce vertical alignment, split too large and too small</comment>
      <origin>2810524</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18158" sha1="fwd950kdw3208q4d2xooj1azbu7j5t3" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These templates are recommended for use by those involved with page deletions:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for large projects and difficult decisions
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts (templates and policy guidelines under construction)

'''Don't worry about selecting the &quot;correct&quot; template''', as more than one template may be appropriate. It is more important to provide assistance by flagging resources that you think shouldn't be on Wikiversity.}}

This '''deletion guideline''' explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources or specific revisions of a resource. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion]].

== Types of deletion ==
=== Speedy ===
Resources may be subject to '''speedy deletion''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources include:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Consider [[Wikiversity:Be bold|boldly]] improving resources and/or discussing alternative solutions such as:

{| class=&quot;wikitable&quot;
! Issue
! Possible alternatives
|-
| valign=&quot;top&quot; |Outside [[WV:scope|Wikiversity's scope]]
| Move to another Wikimedia project or external website as appropriate. Some resources, such as personal essays, may be suitable for userspace.
|-
| valign=&quot;top&quot; |Outside the scope of a learning project
| valign=&quot;top&quot; |Merge into another learning project or use it to start a new learning project.
|-
| valign=&quot;top&quot; |Insufficient or unclear educational value
| valign=&quot;top&quot; |Improve the resource, request clarification, or move it to draft or userspace for further development.
|-
| valign=&quot;top&quot; |Resource is too small
| valign=&quot;top&quot; |Expand or merge small resources into broader resources or redirect them appropriately.
|-
| valign=&quot;top&quot; |Resource is too large
| valign=&quot;top&quot; |Split large resources into smaller resources or reorganise them into subpages.
|-
| valign=&quot;top&quot; |Resource is in a foreign language
| valign=&quot;top&quot; |Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| valign=&quot;top&quot; |Missing or incomplete license information
| valign=&quot;top&quot; |Assist the user in providing the information or supply it if possible.
|-
| valign=&quot;top&quot; |Broken redirects
| valign=&quot;top&quot; |Redirect to a more appropriate target.
|}

==Deletion templates==
To remove resources from mainspace that cannot be directly used by teachers and instructors use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draftspace or userspace.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''
*&lt;code&gt;{{tl|draft archive}}&lt;/code&gt; for proposal for '' moving under Draft:Archive/ '' (proposal is under construction)

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

Note that the deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues.  

'''IMPORTANT:''' The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}
{{Shortcut|Deletion request|Dr|DR|Rfd|RfD}} 

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].
*{{spaces|2}}''See also [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]''

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>fwd950kdw3208q4d2xooj1azbu7j5t3</sha1>
    </revision>
    <revision>
      <id>2810527</id>
      <parentid>2810524</parentid>
      <timestamp>2026-05-20T02:42:04Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Expand and reorganise table of alternatives to deletion with assistance of ChatGPT: https://chatgpt.com/share/6a0d1825-fa38-83ec-89d5-334a0c65eb49</comment>
      <origin>2810527</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19683" sha1="2qbfuzclwtflsj82gw7e5c2asxitc08" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These templates are recommended for use by those involved with page deletions:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for large projects and difficult decisions
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts (templates and policy guidelines under construction)

'''Don't worry about selecting the &quot;correct&quot; template''', as more than one template may be appropriate. It is more important to provide assistance by flagging resources that you think shouldn't be on Wikiversity.}}

This '''deletion guideline''' explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources or specific revisions of a resource. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion]].

== Types of deletion ==
=== Speedy ===
Resources may be subject to '''speedy deletion''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources include:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Participants are encouraged to [[Wikiversity:Be bold|boldly]] consider constructive alternatives to deletion such as: 

{| class=&quot;wikitable&quot;
! Issue
! Possible alternatives
|-
| valign=&quot;top&quot; |Outside [[WV:scope|Wikiversity's scope]]
| valign=&quot;top&quot; |Move to another Wikimedia project or more appropriate external platform. Some resources, such as personal essays, may be suitable for user space.
|-
| valign=&quot;top&quot; |Outside the scope of a learning project
| valign=&quot;top&quot; |Merge into another learning project or develop into a new learning project.
|-
| valign=&quot;top&quot; |Content dispute
| valign=&quot;top&quot; |Seek discussion, mediation, or wider community input before considering deletion.
|-
| valign=&quot;top&quot; |Foreign-language resource
| valign=&quot;top&quot; |Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| valign=&quot;top&quot; |Duplicate resource
| valign=&quot;top&quot; |Merge content, redirect to the primary resource, or move to a subpage with a clearer focus.
|-
| valign=&quot;top&quot; |Poorly organised resource
| valign=&quot;top&quot; |Reorganise sections, rename the resource, or restructure into subpages.
|-
| valign=&quot;top&quot; |Oversized resource
| valign=&quot;top&quot; |Split into smaller resources or reorganise into subpages.
|-
| valign=&quot;top&quot; |Undersized resource
| valign=&quot;top&quot; |Expand, merge into a broader resource, or redirect appropriately.
|-
| valign=&quot;top&quot; |Isolated content
| valign=&quot;top&quot; |Integrate into a broader learning project or create supporting pages.
|-
| valign=&quot;top&quot; |Low-quality content
| valign=&quot;top&quot; |Add cleanup, expansion, or context templates and encourage improvement.
|-
| valign=&quot;top&quot; |Insufficient or unclear educational value
| valign=&quot;top&quot; |Request clarification, improve the resource, or move to draft or user space for further development.
|-
| valign=&quot;top&quot; |Outdated resource
| valign=&quot;top&quot; |Update, archive, or mark as historical or superseded.
|-
| valign=&quot;top&quot; |Inactive learning project
| valign=&quot;top&quot; |Retain, archive, or reorganise if the project continues to have educational value.
|-
| valign=&quot;top&quot; |Abandoned draft
| valign=&quot;top&quot; |Move to draft or user space for future development.
|-
| valign=&quot;top&quot; |Unsuitable resource title
| valign=&quot;top&quot; |Rename or move to a more descriptive title.
|-
| valign=&quot;top&quot; |Unsourced or poorly sourced content
| valign=&quot;top&quot; |Tag for verification, request sources, or assist with adding references.
|-
| valign=&quot;top&quot; |Missing or incomplete licence information
| valign=&quot;top&quot; |Assist the user in providing the information or supply it if possible.
|-
| valign=&quot;top&quot; |Test or practice edits
| valign=&quot;top&quot; |Move to user space or sandbox space if there is constructive or educational value.
|-
| valign=&quot;top&quot; |Broken redirects
| valign=&quot;top&quot; |Redirect to a more appropriate target.
|-
| valign=&quot;top&quot; |Temporary concerns
| valign=&quot;top&quot; |Use maintenance, protection, discussion, or draftification templates until issues are resolved.
|}

==Deletion templates==
To remove resources from mainspace that cannot be directly used by teachers and instructors use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draft space or user space.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''
*&lt;code&gt;{{tl|draft archive}}&lt;/code&gt; for proposal for '' moving under Draft:Archive/ '' (proposal is under construction)

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

Note that the deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues.  

'''IMPORTANT:''' The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}
{{Shortcut|Deletion request|Dr|DR|Rfd|RfD}} 

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].
*{{spaces|2}}''See also [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]''

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>2qbfuzclwtflsj82gw7e5c2asxitc08</sha1>
    </revision>
    <revision>
      <id>2810529</id>
      <parentid>2810527</parentid>
      <timestamp>2026-05-20T02:44:24Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Alternatives to deletion */ Make table sortable</comment>
      <origin>2810529</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19692" sha1="en7tt1774b11pz7byxko660gvp38cnr" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These templates are recommended for use by those involved with page deletions:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for large projects and difficult decisions
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts (templates and policy guidelines under construction)

'''Don't worry about selecting the &quot;correct&quot; template''', as more than one template may be appropriate. It is more important to provide assistance by flagging resources that you think shouldn't be on Wikiversity.}}

This '''deletion guideline''' explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources or specific revisions of a resource. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion]].

== Types of deletion ==
=== Speedy ===
Resources may be subject to '''speedy deletion''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources include:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Participants are encouraged to [[Wikiversity:Be bold|boldly]] consider constructive alternatives to deletion such as: 

{| class=&quot;wikitable sortable&quot;
! Issue
! Possible alternatives
|-
| valign=&quot;top&quot; |Outside [[WV:scope|Wikiversity's scope]]
| valign=&quot;top&quot; |Move to another Wikimedia project or more appropriate external platform. Some resources, such as personal essays, may be suitable for user space.
|-
| valign=&quot;top&quot; |Outside the scope of a learning project
| valign=&quot;top&quot; |Merge into another learning project or develop into a new learning project.
|-
| valign=&quot;top&quot; |Content dispute
| valign=&quot;top&quot; |Seek discussion, mediation, or wider community input before considering deletion.
|-
| valign=&quot;top&quot; |Foreign-language resource
| valign=&quot;top&quot; |Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| valign=&quot;top&quot; |Duplicate resource
| valign=&quot;top&quot; |Merge content, redirect to the primary resource, or move to a subpage with a clearer focus.
|-
| valign=&quot;top&quot; |Poorly organised resource
| valign=&quot;top&quot; |Reorganise sections, rename the resource, or restructure into subpages.
|-
| valign=&quot;top&quot; |Oversized resource
| valign=&quot;top&quot; |Split into smaller resources or reorganise into subpages.
|-
| valign=&quot;top&quot; |Undersized resource
| valign=&quot;top&quot; |Expand, merge into a broader resource, or redirect appropriately.
|-
| valign=&quot;top&quot; |Isolated content
| valign=&quot;top&quot; |Integrate into a broader learning project or create supporting pages.
|-
| valign=&quot;top&quot; |Low-quality content
| valign=&quot;top&quot; |Add cleanup, expansion, or context templates and encourage improvement.
|-
| valign=&quot;top&quot; |Insufficient or unclear educational value
| valign=&quot;top&quot; |Request clarification, improve the resource, or move to draft or user space for further development.
|-
| valign=&quot;top&quot; |Outdated resource
| valign=&quot;top&quot; |Update, archive, or mark as historical or superseded.
|-
| valign=&quot;top&quot; |Inactive learning project
| valign=&quot;top&quot; |Retain, archive, or reorganise if the project continues to have educational value.
|-
| valign=&quot;top&quot; |Abandoned draft
| valign=&quot;top&quot; |Move to draft or user space for future development.
|-
| valign=&quot;top&quot; |Unsuitable resource title
| valign=&quot;top&quot; |Rename or move to a more descriptive title.
|-
| valign=&quot;top&quot; |Unsourced or poorly sourced content
| valign=&quot;top&quot; |Tag for verification, request sources, or assist with adding references.
|-
| valign=&quot;top&quot; |Missing or incomplete licence information
| valign=&quot;top&quot; |Assist the user in providing the information or supply it if possible.
|-
| valign=&quot;top&quot; |Test or practice edits
| valign=&quot;top&quot; |Move to user space or sandbox space if there is constructive or educational value.
|-
| valign=&quot;top&quot; |Broken redirects
| valign=&quot;top&quot; |Redirect to a more appropriate target.
|-
| valign=&quot;top&quot; |Temporary concerns
| valign=&quot;top&quot; |Use maintenance, protection, discussion, or draftification templates until issues are resolved.
|}

==Deletion templates==
To remove resources from mainspace that cannot be directly used by teachers and instructors use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draft space or user space.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''
*&lt;code&gt;{{tl|draft archive}}&lt;/code&gt; for proposal for '' moving under Draft:Archive/ '' (proposal is under construction)

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

Note that the deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues.  

'''IMPORTANT:''' The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}
{{Shortcut|Deletion request|Dr|DR|Rfd|RfD}} 

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].
*{{spaces|2}}''See also [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]''

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>en7tt1774b11pz7byxko660gvp38cnr</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Please do not bite the newcomers</title>
    <ns>4</ns>
    <id>5596</id>
    <revision>
      <id>2810535</id>
      <parentid>2128646</parentid>
      <timestamp>2026-05-20T03:04:43Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Replace missing image</comment>
      <origin>2810535</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2052" sha1="0cf42a9934f8o5g7r0ujtqkd0i7m2h8" xml:space="preserve">{{proposal|WV:BITE}}

Please avoid biting newcomers.
[[File:Playing_with_the_ball_3.jpg|frame|right|Newcomers may seem delicious, but please don't bite them!]]
==On experience==
Experience comes with time. Too often online collaborations turn into social cliques where outsiders are disparaged, and those not already in the know are actively discouraged from participating. When you see a new post on an old topic, don't attack the poster as ignorant, naive or of dubious parentage.  Simply direct them to the information they are seeking. Treat each new member with great respect. Each new member offers new knowledge, skills, and specialties. Unless they are obviously trying to be malicious, give them a link to a page that offers help, and don’t discourage them.

==More on newcomers==
Newcomers and even experienced Wikiversitians may not be familiar with the [[Wikiversity:Policies|policies]], scope, or [[Wikiversity:Mission|mission of Wikiversity]]. If someone acting in good faith violates a policy, acts outside the scope, or acts counter productively to the mission of Wikiversity, please do not bite them. Please be sensitive, empathetic, tactful, and considerate when letting a user know there is a more standard, accepted, or &quot;proper&quot; way to do something. Sometimes it is best to let a user know about a policy, or tradition. Sometimes just fixing it yourself and not saying anything is the best course of action, and sometimes it is best to just let the user be and let them figure it out on their own. One should use discernment and good judgment in this regard.

==Quotes==
Everyone at Wikiversity is a newcomer. We are all exploring how to use wiki technology in support of education. We need a &quot;culture of thoughtful, diplomatic honesty... Diplomacy consists of combining honesty and politeness.&quot; --[[w:User:Jimbo Wales/Statement of principles|Jimbo Wales]] (December 17, 2006)

==See also==
* [[Wikiversity:Assume Good Faith]]
* [[Managing vandalism]]

{{official policies}}
{{proposed policies}}

[[Category:Wikiversity culture]]</text>
      <sha1>0cf42a9934f8o5g7r0ujtqkd0i7m2h8</sha1>
    </revision>
    <revision>
      <id>2810536</id>
      <parentid>2810535</parentid>
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      <text bytes="2058" sha1="7rwfla70s2sxrlwu01rixcvbq7ybqr3" xml:space="preserve">{{proposal|WV:BITE}}

Please avoid biting newcomers.
[[File:Playing_with_the_ball_3.jpg|frame|right|225px|Newcomers may seem delicious, but please don't bite them!]]
==On experience==
Experience comes with time. Too often online collaborations turn into social cliques where outsiders are disparaged, and those not already in the know are actively discouraged from participating. When you see a new post on an old topic, don't attack the poster as ignorant, naive or of dubious parentage.  Simply direct them to the information they are seeking. Treat each new member with great respect. Each new member offers new knowledge, skills, and specialties. Unless they are obviously trying to be malicious, give them a link to a page that offers help, and don’t discourage them.

==More on newcomers==
Newcomers and even experienced Wikiversitians may not be familiar with the [[Wikiversity:Policies|policies]], scope, or [[Wikiversity:Mission|mission of Wikiversity]]. If someone acting in good faith violates a policy, acts outside the scope, or acts counter productively to the mission of Wikiversity, please do not bite them. Please be sensitive, empathetic, tactful, and considerate when letting a user know there is a more standard, accepted, or &quot;proper&quot; way to do something. Sometimes it is best to let a user know about a policy, or tradition. Sometimes just fixing it yourself and not saying anything is the best course of action, and sometimes it is best to just let the user be and let them figure it out on their own. One should use discernment and good judgment in this regard.

==Quotes==
Everyone at Wikiversity is a newcomer. We are all exploring how to use wiki technology in support of education. We need a &quot;culture of thoughtful, diplomatic honesty... Diplomacy consists of combining honesty and politeness.&quot; --[[w:User:Jimbo Wales/Statement of principles|Jimbo Wales]] (December 17, 2006)

==See also==
* [[Wikiversity:Assume Good Faith]]
* [[Managing vandalism]]

{{official policies}}
{{proposed policies}}

[[Category:Wikiversity culture]]</text>
      <sha1>7rwfla70s2sxrlwu01rixcvbq7ybqr3</sha1>
    </revision>
    <revision>
      <id>2810537</id>
      <parentid>2810536</parentid>
      <timestamp>2026-05-20T03:05:49Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <origin>2810537</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2058" sha1="35hh9iwsi4mqsem6tjigqg9g33tt5vm" xml:space="preserve">{{proposal|WV:BITE}}

Please avoid biting newcomers.
[[File:Playing_with_the_ball_3.jpg|right|255px|thumb|Newcomers may seem delicious, but please don't bite them!]]
==On experience==
Experience comes with time. Too often online collaborations turn into social cliques where outsiders are disparaged, and those not already in the know are actively discouraged from participating. When you see a new post on an old topic, don't attack the poster as ignorant, naive or of dubious parentage.  Simply direct them to the information they are seeking. Treat each new member with great respect. Each new member offers new knowledge, skills, and specialties. Unless they are obviously trying to be malicious, give them a link to a page that offers help, and don’t discourage them.

==More on newcomers==
Newcomers and even experienced Wikiversitians may not be familiar with the [[Wikiversity:Policies|policies]], scope, or [[Wikiversity:Mission|mission of Wikiversity]]. If someone acting in good faith violates a policy, acts outside the scope, or acts counter productively to the mission of Wikiversity, please do not bite them. Please be sensitive, empathetic, tactful, and considerate when letting a user know there is a more standard, accepted, or &quot;proper&quot; way to do something. Sometimes it is best to let a user know about a policy, or tradition. Sometimes just fixing it yourself and not saying anything is the best course of action, and sometimes it is best to just let the user be and let them figure it out on their own. One should use discernment and good judgment in this regard.

==Quotes==
Everyone at Wikiversity is a newcomer. We are all exploring how to use wiki technology in support of education. We need a &quot;culture of thoughtful, diplomatic honesty... Diplomacy consists of combining honesty and politeness.&quot; --[[w:User:Jimbo Wales/Statement of principles|Jimbo Wales]] (December 17, 2006)

==See also==
* [[Wikiversity:Assume Good Faith]]
* [[Managing vandalism]]

{{official policies}}
{{proposed policies}}

[[Category:Wikiversity culture]]</text>
      <sha1>35hh9iwsi4mqsem6tjigqg9g33tt5vm</sha1>
    </revision>
  </page>
  <page>
    <title>User:Juandev</title>
    <ns>2</ns>
    <id>8453</id>
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      <timestamp>2026-05-20T10:45:12Z</timestamp>
      <contributor>
        <username>Juandev</username>
        <id>2651</id>
      </contributor>
      <comment>sick time</comment>
      <origin>2810582</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1065" sha1="jerank84sce0a0ojgeyw2hgdaw7nme7" xml:space="preserve">'''I am sick, so I'll be less around these days.'''{{userboxtop}}
{{User custodian}}
{{userboxbottom}}
{{Babel|cs|en-3|es-3|pl-1}}
I graduated from the Czech University of Life Sciences in Prague and studied information science at the Faculty of Arts of Charles University. In 2005, I started contributing to Wikipedia for the first time. Since then, I've been active on Wikipedia, Wikimedia Commons, Wikiversity, and Wikidata, contributing content and organizing community and outreach projects. I have primarily contributed to Wikiversity in the Czech language in recent years, which I created in 2007. 

=== What am I doing ===

==== Policies and guidelines ====

* working on policy and guidelines proposals to prepare a draft of [[User:Juandev/Making policy|overarching policy]]
* studying Copyrights pages to propose a new Wikiversity:Copyrights draft
* t[[User:Juandev/W:Artificial intelligence/notes|hinking on artificial inteligence usage]] on to contribute to the proposed policy
* hope to propose Wikiversity:Etics in the future

[[cs:Uživatel:Juandev]]</text>
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    <title>Wikiversity:Bots/item</title>
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        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <minor />
      <comment>Protected &quot;[[Wikiversity:Bots/item]]&quot;: Highly visible template ([Edit=Allow only autoconfirmed users] (indefinite) [Move=Allow only autoconfirmed users] (indefinite))</comment>
      <origin>1502571</origin>
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      <text bytes="302" sha1="sz6m7f1qqh3lfq55ud95mdcn31uei37" xml:space="preserve">&lt;noinclude&gt;{| border=&quot;0&quot; align=&quot;center&quot; rules=&quot;all&quot; cellpadding=&quot;3px&quot; class=&quot;wikitable&quot;
!Name of the bot
!Contributions
!Operator
!Functions
!Last activity
&lt;/noinclude&gt;|-
|[[User:{{{1}}}]]
|[[Special:Contributions/{{{1}}}|Contribs]]
|[[User:{{{2}}}|{{{2}}}]]
|{{{3}}}
|{{{4}}}
&lt;noinclude&gt;|}&lt;/noinclude&gt;</text>
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  <page>
    <title>Portal:Structural biology</title>
    <ns>102</ns>
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      <id>2810459</id>
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'''Welcome to the Department of Structural Biology!'''
{{welcome}}
'''Structural biology''' is a branch of [[molecular biology]] concerned with the study of the architecture and shape of biological macromolecules—proteins and nucleic acids in particular—and what causes them to have the structures they have. This subject is of great interest to biologists, because macromolecules carry out most of the functions of a cell, and because typically it only is by coiling into a specific three-dimensional shape that they are able to perform their functions. This shape, which is called the &quot;tertiary structure&quot; of a molecule, depends in a complicated way on the molecule's basic composition, or &quot;primary structure.&quot;

Biomolecules are too small to see in detail even with the most advanced light microscopes. The methods that structural biologists use to determine their structures generally involve measurements on vast numbers of identical molecules at the same time. These methods include crystallography, NMR, ultra fast laser spectroscopy, electron microscopy, electron cryomicroscopy (cryo-EM), and circular dichroism. Most often researchers use them to study the static &quot;native states&quot; of macromolecules. But variations on these methods are also used to watch nascent or denatured molecules assume or reassume their native states (see e.g. protein folding).

A third approach that structural biologists take to understanding structure is bioinformatics to look for patterns among the diverse sequences that give rise to particular shapes. Researchers often can deduce aspects of the structure of integral membrane proteins based on the membrane topology predicted by hydrophobicity analysis. See: protein structure prediction.

In the past few years it has become possible for highly accurate physical molecular models to complement the in silico study of biological structures. Rapid prototyping technologies such as those used by 3D Molecular Design, or the creation of molecular models in glass by Luminorum Ltd, are notable examples of recent advances in this field.

== Department news ==
*'''February 28, 2007''' - Department founded!

== Learning projects ==

== Research ==

== Active participants ==

== Wikibook textbooks ==

[[Category:Biology|*]]
&lt;u&gt;The Structure of Life&lt;/u&gt; [http://publications.nigms.nih.gov/structlife/structlife.pdf]Introductory Booklet by the US Department of Health and Human Services

== External links ==
* [https://cryoemservices.com CryoEM Services] – A national directory and searchable database of cryo-electron microscopy (cryo-EM) core facilities, instruments, and related services for researchers.</text>
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      <timestamp>2026-05-19T16:36:52Z</timestamp>
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        <id>276019</id>
      </contributor>
      <minor />
      <comment>Reverted edit by [[Special:Contributions/~2026-30174-57|~2026-30174-57]] ([[User_talk:~2026-30174-57|talk]]) to last version by [[User:Hasley|Hasley]] using [[Wikiversity:Rollback|rollback]]</comment>
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'''Welcome to the Department of Structural Biology!'''
{{welcome}}
'''Structural biology''' is a branch of [[molecular biology]] concerned with the study of the architecture and shape of biological macromolecules—proteins and nucleic acids in particular—and what causes them to have the structures they have. This subject is of great interest to biologists, because macromolecules carry out most of the functions of a cell, and because typically it only is by coiling into a specific three-dimensional shape that they are able to perform their functions. This shape, which is called the &quot;tertiary structure&quot; of a molecule, depends in a complicated way on the molecule's basic composition, or &quot;primary structure.&quot;

Biomolecules are too small to see in detail even with the most advanced light microscopes. The methods that structural biologists use to determine their structures generally involve measurements on vast numbers of identical molecules at the same time. These methods include crystallography, NMR, ultra fast laser spectroscopy, electron microscopy, electron cryomicroscopy (cryo-EM), and circular dichroism. Most often researchers use them to study the static &quot;native states&quot; of macromolecules. But variations on these methods are also used to watch nascent or denatured molecules assume or reassume their native states (see e.g. protein folding).

A third approach that structural biologists take to understanding structure is bioinformatics to look for patterns among the diverse sequences that give rise to particular shapes. Researchers often can deduce aspects of the structure of integral membrane proteins based on the membrane topology predicted by hydrophobicity analysis. See: protein structure prediction.

In the past few years it has become possible for highly accurate physical molecular models to complement the in silico study of biological structures. Rapid prototyping technologies such as those used by 3D Molecular Design, or the creation of molecular models in glass by Luminorum Ltd, are notable examples of recent advances in this field.

== Department news ==
*'''February 28, 2007''' - Department founded!

== Learning projects ==

== Research ==

== Active participants ==

== Wikibook textbooks ==

[[Category:Biology|*]]
&lt;u&gt;The Structure of Life&lt;/u&gt; [http://publications.nigms.nih.gov/structlife/structlife.pdf]Introductory Booklet by the US Department of Health and Human Services</text>
      <sha1>mk4aa9g5cpdxdwem1qumfocereoz0mk</sha1>
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  </page>
  <page>
    <title>Film writing</title>
    <ns>0</ns>
    <id>31520</id>
    <revision>
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      <timestamp>2026-05-19T13:04:47Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>RFD</comment>
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==Film Writing==

Any Writing fulfilling the technical needs for making a film can be called film writing.

This is not the same as normal writing.

The film writing is classified in the following catagories:

Concept

Screenplay

Dialogues

===Concept===

This is a basic story line for a film. This contains character mapping i.e. detailed description of each character and story line with various scenes expected in the film.

===Screenplay===

This is a series of scenes described as they must be shot in a film. This contains the details expected in every scene as per the scenes narrated in the story line with details of characters present in it..

===Dialogues===

This is the last stage of film writing where the dialogue writer writes the dialogues for each character.

===Brainstorming===
Interested in brainstorming ideas for screen plays? This can be good practice and learning for you, and a potential starting point for someone interested in learning to develop a screenplay. Here at Wikiversity, we learn by doing.

* [[/Brainstorming/]]
* [[/Brainstorming/Wicklow Games]]

===Active participants===
[[User:Rekhaa Kale|Rekhaa Kale]] 19:56, 22 March 2007 (UTC)

[[Category:Film|writing]]
[[Category:Writing]]</text>
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      <comment>Change to more specific category</comment>
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==Film Writing==

Any Writing fulfilling the technical needs for making a film can be called film writing.

This is not the same as normal writing.

The film writing is classified in the following catagories:

Concept

Screenplay

Dialogues

===Concept===

This is a basic story line for a film. This contains character mapping i.e. detailed description of each character and story line with various scenes expected in the film.

===Screenplay===

This is a series of scenes described as they must be shot in a film. This contains the details expected in every scene as per the scenes narrated in the story line with details of characters present in it..

===Dialogues===

This is the last stage of film writing where the dialogue writer writes the dialogues for each character.

===Brainstorming===
Interested in brainstorming ideas for screen plays? This can be good practice and learning for you, and a potential starting point for someone interested in learning to develop a screenplay. Here at Wikiversity, we learn by doing.

* [[/Brainstorming/]]
* [[/Brainstorming/Wicklow Games]]

===Active participants===
[[User:Rekhaa Kale|Rekhaa Kale]] 19:56, 22 March 2007 (UTC)

[[Category:Filmmaking]]
[[Category:Writing]]</text>
      <sha1>mtryjc2yexf3w81edqvd5tbzv1w6lex</sha1>
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    <revision>
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        <username>Jtneill</username>
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      <comment>/* Film Writing */ Copyedit</comment>
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Any writing fulfilling the technical needs for making a film can be called film writing.

This is not the same as normal writing.

The film writing is classified in the following catagories:
* Concept
* Screenplay
* Dialogues

==Concept==
This is a basic story line for a film. This contains character mapping i.e. detailed description of each character and story line with various scenes expected in the film.

==Screenplay==
This is a series of scenes described as they must be shot in a film. This contains the details expected in every scene as per the scenes narrated in the story line with details of characters present in it..

==Dialogues==
This is the last stage of film writing where the dialogue writer writes the dialogues for each character.

==Brainstorming==
Interested in brainstorming ideas for screen plays? This can be good practice and learning for you, and a potential starting point for someone interested in learning to develop a screenplay. Here at Wikiversity, we learn by doing.

* [[/Brainstorming/]]
* [[/Brainstorming/Wicklow Games]]

===Active participants===
[[User:Rekhaa Kale|Rekhaa Kale]] 19:56, 22 March 2007 (UTC)

[[Category:Filmmaking]]
[[Category:Writing]]</text>
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      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Move brainstorming to top and update</comment>
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Any writing fulfilling the technical needs for making a film can be called film writing.

This is not the same as normal writing.

The film writing is classified in the following catagories:
* Concept
* Screenplay
* Dialogues

==Brainstorming==
[[w:Brainstorm|Brainstorm]] ideas for screen plays. This can be good practice and learning, and a potential starting point for learning to develop a screenplay.

==Concept==
This is a basic story line for a film. This contains character mapping (i.e. detailed description of each character and story line with various scenes expected in the film).

==Screenplay==
This is a series of scenes described as they must be shot in a film. This contains the details expected in every scene as per the scenes narrated in the story line with details of characters present in it.

==Dialogues==
This is the last stage of film writing where the dialogue writer writes the dialogues for each character.

==Next step==
Proceed to [[filmmaking]].

[[Category:Filmmaking]]
[[Category:Writing]]</text>
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    <revision>
      <id>2810533</id>
      <parentid>2810532</parentid>
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        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
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Any writing fulfilling the technical needs for making a film can be called film writing.

This is not the same as normal writing.

The film writing process can be oganised into the following catagories:
* Brainstorming
* Concept
* Screenplay
* Dialogues

==Brainstorming==
[[w:Brainstorm|Brainstorm]] ideas for screen plays. This can be good practice and learning, and a potential starting point for learning to develop a screenplay.

==Concept==
This is a basic story line for a film. This contains character mapping (i.e. detailed description of each character and story line with various scenes expected in the film).

==Screenplay==
This is a series of scenes described as they must be shot in a film. This contains the details expected in every scene as per the scenes narrated in the story line with details of characters present in it.

==Dialogues==
This is the last stage of film writing where the dialogue writer writes the dialogues for each character.

==Next step==
Proceed to [[filmmaking]].

[[Category:Filmmaking]]
[[Category:Writing]]</text>
      <sha1>pct344rdlhzht86kij1323oedvj3ohg</sha1>
    </revision>
    <revision>
      <id>2810538</id>
      <parentid>2810533</parentid>
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        <username>Jtneill</username>
        <id>10242</id>
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Any writing fulfilling the technical needs for making a film can be called film writing.

This is not the same as normal writing.

The film writing process can be oganised into the following steps:
* Brainstorming
* Concept
* Screenplay
* Dialogues

==Brainstorming==
[[w:Brainstorm|Brainstorm]] ideas for screen plays. This can be good practice and learning, and a potential starting point for learning to develop a screenplay.

==Concept==
This is a basic story line for a film. This contains character mapping (i.e. detailed description of each character and story line with various scenes expected in the film).

==Screenplay==
This is a series of scenes described as they must be shot in a film. This contains the details expected in every scene as per the scenes narrated in the story line with details of characters present in it.

==Dialogues==
This is the last stage of film writing where the dialogue writer writes the dialogues for each character.

==Next step==
Proceed to [[filmmaking]].

[[Category:Filmmaking]]
[[Category:Writing]]</text>
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  </page>
  <page>
    <title>Solar energy</title>
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    <revision>
      <id>2810510</id>
      <parentid>2762215</parentid>
      <timestamp>2026-05-19T23:46:22Z</timestamp>
      <contributor>
        <username>IanVG</username>
        <id>2918363</id>
      </contributor>
      <comment>added equations</comment>
      <origin>2810510</origin>
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      <format>text/x-wiki</format>
      <text bytes="16487" sha1="pgh24gwqt0qaj4pgor56h7x8t6cbinc" xml:space="preserve">[[Image:Solar concentrator.jpg|thumb|right|300px|[[Topic:Solar energy|Solar energy]] can be concentrated using a parabolic reflector to achieve very high temperatures.]]

Welcome to the topic of solar energy! In this topic area you will find courses devoted to the engineering and science of solar energy, and its diverse applications and systems. Please see the page on [[renewable energy]] for other renewable energy applications.

Solar energy systems come in a diverse range of technologies and are largely divided up into two main categories which are solar electric and solar heating. Solar electric applications covers solar photovoltaic cells used for generating electricity, while solar heating systems typically are used for heating water.

A further subdivision could be thought of as '''active''' and '''passive''' systems. Active systems are those which are built solely to capture solar energy whereas as passive systems are more generally design features that make use of the available solar energy. An example of the latter is building architecture.

== Courses ==
[[Introduction to solar energy]]

==The Resource==
[[Solar energy/Resource availability]]

At the top of the atmosphere the solar irradiance on the Earth is 1,366 watts per square meter (w/m-&lt;sup&gt;2&lt;/sup&gt;).&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=y_GMTXRtxJ8C&amp;newbks=0&amp;printsec=frontcover&amp;pg=PA36&amp;dq=At+the+top+of+the+atmosphere+the+solar+irradiance+on+the+Earth+is+1,366+watts+per+square+meter+(w/m-2).&amp;hl=en|title=Climate Change: Myths and Realities|publisher=Jeevananda Reddy|language=en}}&lt;/ref&gt; This means that the total solar energy received by the Earth is: 

&lt;math&gt;\pi \times \text {radius}_{earth}^2 \times 1366 \frac{W}{m^2} &lt;/math&gt; 

Plugging in the values this gives:

 3.14159 * 6,378,100 * 6,378,100 * 1,366 = 174,575,322,671,459,743 watts 

or rounding off this number it is: 1.74 * 10&lt;sup&gt;17&lt;/sup&gt; watts or 170,000 TWh where a terrawatt is 1 * 10&lt;sup&gt;12&lt;/sup&gt; watts

Clearly this value is not exact because the Earth is in an elliptic orbit and so the amount of solar energy varies slightly throughout the year and also the Earth bulges ever so slightly. But these effects will not change the overall magnitude of the figure. 

What matters though is the amount of solar energy hitting the ground at any point and this varies depending on the latitude, the season, the local time and the thickness of the atmosphere that it has to travel through. In general at least 30% of the 1.36 kWh is lost in the atmosphere while at higher latitudes where the sunlight has to pass through more atmosphere because it is entering at an angle it can probably be as high as 50%. The effect of season is most pronounced at higher latitudes.

A point to note about local time, is that mid-day local time is generally defined as the time at which the sun reaches the highest point in the sky at that location. In reality then there is just a few hours in the day when the sun is around it's maximum angular height in the sky and supplying most energy

==The Vision==
It has long been the dream of many solar proponents that solar energy can provide all our energy needs. In 2004, the '''worldwide energy consumption''' of the human race was on average 15 TW (= 1.5 x 10&lt;sup&gt;13&lt;/sup&gt; W).&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=xqV4Gs4hKoEC&amp;newbks=0&amp;printsec=frontcover&amp;pg=PA9&amp;dq=worldwide+energy+consumption+&amp;hl=en|title=Energy Efficiency and Human Activity: Past Trends, Future Prospects|last=Schipper|first=Lee|last2=Meyers|first2=Stephen|date=1992-11-19|publisher=Cambridge University Press|isbn=978-0-521-43297-9|language=en}}&lt;/ref&gt; The figures above show that the Earth receives more than 10,000 times this amount of energy and show clearly there is enough to provide all our energy needs.

It is hard to argue with these figures and it has been suggested many times that the one off gift of nature of fossil fuel energy should be used to build the solar energy infrastructure which can last humankind indefinitely given that other problems like ecological collapse, global warming and a pollution crisis are avoided.

A popular calculation that has been done is to determine how many square kilometers would need to be covered by solar energy to capture enough energy for our needs. The figure widely quoted is a box about 100 miles (or 160 km) per side which is 10,000 square miles (25,600 km&lt;sup&gt;2&lt;/sup&gt;) but this figure is just for the United States.

The calculation is derived roughly as follows. It is assumed that the location is at a relatively low latitude in sunny place like a desert and that 1000 w/m&lt;sup&gt;2&lt;/sup&gt; are received at ground level. Taking current solar photovoltaic cell efficiencies actually deployed, we assume a value of around 10% efficiency. Therefore to achieve 15 Twh (10&lt;sup&gt;12&lt;/sup&gt;w):

  Area = 15 Twh / 1000 w/m&lt;sup&gt;2&lt;/sup&gt; * 0.1 = 1.5 * 10&lt;sup&gt;11&lt;/sup&gt; m&lt;sup&gt;2&lt;/sup&gt;

Taking the square root and converting from m&lt;sup&gt;2&lt;/sup&gt; to km&lt;sup&gt;2&lt;/sup&gt; gives:

387,298 metres or '''387 km''' per side. Doing the calculation using an efficiency of 15% instead of 10% gives a value of '''316 km''' (~197 miles) which is considerably lower. And an even higher efficiency of 20% which would be less likely for such a large system would reduce this to '''274 km'''.

If this calculation was done for just the total global amount of electricity used then it would be lower. Other factors to consider are that by introducing more efficient energy technologies and greatly increasing the amount of public transport would greatly reduce the amount of oil and thereby total energy usage.

==The Needs==

=Technologoies=
[[Solar Energy Technologies|Solar Energy Technologies]]

==Solar Electric==
The generation of electricity from solar energy is dominated by the use of solar photovoltaic cells. However there is the less well known solar thermovoltaic effect which uses heat instead to convert solar infrared light into electricity directly.
===Solar [[Photovoltaics]]===
These convert sunlight into directly electricity. For a full discussion of this process see [[[w:Solar_Cell]]].

There are three main types of solar cells. These are '''crystalline silicon solar cells, amphorous silicon solar cells''' and '''thin film solar cells'''.

====Crystalline Solar Cells====
These were the first types of solar cells and require a very pure silicon crystal to be grown from which the solar cell is manufactured. The disadvanages are that the crystals are time consuming and require a lot of energy to grow them. This is because the silicon is held at a high temperature. Indeed this is one of the reasons why it has been so difficult to rapidly increase global production of crystalline solar cells.

As a result of this, the energy payback time of crystalline solar cells has been typically 10 to 20 years, although this is being reduced slowly as the technologies improves and the cells are being manufactured using thin slices of silicon.

One of the main advantages though of crystalline cells is that they tend to have the highest efficiencies of all solars and over the past decade or so, figures from 25% to just above 30% have been achieved. Typically values in the field are less.

====Amorphous Solar Cells====

====Thin Film Solar Cells====
Thin Film solar cells cover a diverse range of cells types using a number of different chemical elements. For a full discussion of this see [[w:Solar_Cell#Thin_films]].

The main advantages of thin film cells is that their energy payback times tend to be lower and they tend to be much more flexible and less brittle. Some types can be even bent and rolled up as sheets.

Thin film cells currently cover about 10% or less of total global production of solar cells.

One of the less discussed problems with thin film solar cells is that they tend to use both relatively rare and toxic chemicals which could present longer term environmental problems.

====Concentrating Systems====
Silicon crystalline solar cells can be used in conjunction with solar concentrates to increase the amount of sunlight following on a given cell. Some types of solar cells even achieve slightly higher efficiencies by doing this.

The main advantages of employing this technique is that you can use much more expensive and higher effeciency cells and at the same time use collect a bigger area of sunlight using cheaper components and thereby reducing your overall cost.

===Solar Thermovoltaics===
These type of solar cells operate by converting infrared light into electricity. They are effectively the same as solar photovoltaics except they are just operating at a longer wavelength. The sun produces a significant amount of infrared radiation as to any hot objects like furnaces. In theory these cells can also be used to operate off the infrared emitted by furnaces.

===Solar Tower===
These are still largely in the experimental phase. They are often referred to as solar updraft tower because they consist of a canopy over a relatively large area under which the area is heated. The canopy is angled upward towards the base of the tower and this allows the heated air to flow under convention up the tower. Significant updraft is created and this can be enough to drive a turbine in the tower shaft.

===Solar Stirling Engine===

==Solar Heating and Cooling==
In addition to transmitting energy in the visible spectrum, the Sun transmits energy in the infrared spectrum. Because materials such as water and metal are good at absorbing wavelengths in this spectrum, solar thermal presents an opportunity to reduce water and space heating needs by supplementing them with solar energy. 

Solar Heating can be further classified into active and passive systems. Examples of active systems are solar water panels that you often see on roofs for heating water, whereas a south facing building that takes advantage of the sun can be considered as a passive form of solar energy.

Solar heating and cooling is the less glamorous side of solar energy compared to the more high tech solar photovoltaics. However it is an important area because significant amounts of energy are used for heating water and for heating and cooling buildings. In the USA, electrical demand is actually higher in the summer rather than the winter because of the amount of electrical powered air-conditioning use. There are considerable opportunities for replacing and reducing the amount of fossil fuel energy to supply these services.

===Solar Panels===
Solar panels are now used quite extensively domestically especially in Mediterranean countries and Israel for heating water.

They basically consist of a radiator painted black to absorb the sunlight and covered by glass to trap the heat. The water is circulated through the narrow piping so that it has a relatively long distance to travel and therefore gives it time to pick up the heat.

At present there is a huge range of panel designs and manufacturers. Solar panels are the ultimate for the do it yourself (DIY) enthusiast because anyone with basic plumbing skills should be able to put together a relatively simple system. Vacuum tube seems to offer the best solution as a reflector is placed under them and this enables 360 Degree sunlight around the tube effectively using the whole tube while the sun shines. Temperature of 303 Degree Celsius has been measured of the inner copper tip which is used on high pressure systems. The low pressure system has no inner copper tube however temperature of 75 degree has been measured. The time is now to make use of natural elements and eliminate destroying the planet. Lets use this technology, its free.

===Solar Chimney===
A solar chimney is a way of improving the natural ventilation of buildings by using convection of air heated by the sun.

See [[w:Solar_chimney]]

==Passive Solar Energy==

===Passive Solar Heating and Cooling===

====Building Orientation====

===Passive Solar Cooling===

====Thick Walls====

===Passive Solar Lighting===

====Sky Lights====

====Solar Tubes====

=Generation of Electricity=
[[Image:Dye sensitized cell phys.jpg|300px|thumb|Dye sensitized solar cell]]
[[Generation of Electricity by Solar|Generation of Electricity by Solar]]

==Utility Factor==
The utility factor refers to the fraction of time in a day that can be used to generate solar power.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=udxvAAAAQBAJ&amp;newbks=0&amp;printsec=frontcover&amp;q=The+utility+factor+refers+to+the+fraction+of+time+in+a+day+that+can+be+used+to+generate+solar+power.&amp;hl=en|title=Solar Energy: Technologies and Project Delivery for Buildings|last=Walker|first=Andy|date=2013-08-07|publisher=John Wiley &amp; Sons|isbn=978-1-118-41654-9|language=en}}&lt;/ref&gt; Clearly average over a year, it is going to be a theoretical maximum of only 50%. In practice this value will be even lower because if you account for the period around both sunrise and sunset when the sun is at a very low angle in the sky, there will not be sufficient power from the available light to generate any significant amounts of power. Taking this into consideration then would suggest a value of 40% being more typical.

It might be worth noting that the typical utility factors achieved for wind power so far are around 26% on average with the best sites as high as 35%.

Utility factors for fossil fuel power stations burning coal, gas, oil and nuclear power stations can be as high as 80% to 90% since other than maintenance there are no other limitations. In situations of a severe drought though, the availability of fresh water for cooling can impact this.

Therefore to replace a given electricial capacity of a fossil fuel power station typically requires double the capacity when being replaced by a solar electrical plant. So for example to replace a 500 MW coal plant would require an installation of 1000 MW (1GW) of solar.

===Capacity and Energy Usage===
When discussing any energy supply it is important to always distinguish between figures for installed capacity and actual energy produced. One should look out for units of wattage which are usually quoted for capacity and kilowatt hours (Kwh) which are used for power produced.

In a traditional fossil fuel power station, it typically might run at close to full power at its utility factor. Its yearly power output will then be approximately:

   365 days * 24 hours * 0.8 Utility Factor * Power Station Capacity (MW)

'''For a 500 MW power station this is:  365 * 24 * 0.8 * 500 = 3,504,000 MWh =''' 3.5 TwH

A figure worth remembering is that there are approximately '''8,760 hours per year'''.

==Residential Power==
The generation of electricity by solar photovoltaic for residential use has been growing reasonably steadily over the years. It is still at a very small level, certainly far less than 1% of domestic electrical consumption. 

For a residential setting the quantity of power needed varies but it ranges from about 2Kw to 4Kw systems. They can be setup as '''grid connected''' or '''off-grid'''. For off-grid systems a battery bank is needed for storing power and this can add to the cost.

==Commercial Power==

==Utility Scale Power==

==Open educational resources==

* [https://ecampusontario.pressbooks.pub/solarpv/ Solar Photovoltaics for Design Engineers]

===Wikipedia===
* [[w:Dye-sensitized solar cells|Dye-sensitized solar cells]]
* [[w:List of photovoltaics companies|List of photovoltaics companies]]
* [[w:Agrivoltaics|Agrivoltaics]]

== Tables of solar energy equations ==

=== Solar thermal equations ===
Useful thermal heat output:

&lt;math&gt;\dot Q''_u = \frac {\dot m_f \cdot C_{p,f} \cdot (T_{out} - T_{in})}{A}
&lt;/math&gt;

Instantaneous thermal efficiency of a solar thermal collector:

&lt;math&gt;\eta_c = \frac {\dot Q''_u}{\dot G''}&lt;/math&gt;

==See also==
[[Renewable energy/Solar thermal]]

== External links ==
* Youtube: [http://www.youtube.com/watch?v=qYeynLy6pj8 How solar panels are assembled]
* [http://www.appropedia.org/Category:Photovoltaics Appropedia:Category:Photovoltaics]
* September 2008 ''[http://www.nss.org/news/releases/pr20080909.html First-of-a-Kind Long-Distance Demonstration of Solar-Powered Wireless Power Transmission Technology]''
* June 2008 ''[http://www.eurekalert.org/pub_releases/2008-06/epfd-neb062708.php New efficiency benchmark for dye-sensitized solar cells]''
* October 2008 ''[http://www.makeitrightnola.org/ Brad Pitt's green building project in New Orlean's Lower 9th Ward]''

[[Category:Renewable energy]]

{{BookCat}}</text>
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      <text bytes="16815" sha1="474atqqtu4n87fthdlf8wejtmxh90xp" xml:space="preserve">[[Image:Solar concentrator.jpg|thumb|right|300px|[[Topic:Solar energy|Solar energy]] can be concentrated using a parabolic reflector to achieve very high temperatures.]]

Welcome to the topic of solar energy! In this topic area you will find courses devoted to the engineering and science of solar energy, and its diverse applications and systems. Please see the page on [[renewable energy]] for other renewable energy applications.

Solar energy systems come in a diverse range of technologies and are largely divided up into two main categories which are solar electric and solar heating. Solar electric applications covers solar photovoltaic cells used for generating electricity, while solar heating systems typically are used for heating water.

A further subdivision could be thought of as '''active''' and '''passive''' systems. Active systems are those which are built solely to capture solar energy whereas as passive systems are more generally design features that make use of the available solar energy. An example of the latter is building architecture.

== Courses ==
[[Introduction to solar energy]]

==The Resource==
[[Solar energy/Resource availability]]

At the top of the atmosphere the solar irradiance on the Earth is 1,366 watts per square meter (w/m-&lt;sup&gt;2&lt;/sup&gt;).&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=y_GMTXRtxJ8C&amp;newbks=0&amp;printsec=frontcover&amp;pg=PA36&amp;dq=At+the+top+of+the+atmosphere+the+solar+irradiance+on+the+Earth+is+1,366+watts+per+square+meter+(w/m-2).&amp;hl=en|title=Climate Change: Myths and Realities|publisher=Jeevananda Reddy|language=en}}&lt;/ref&gt; This means that the total solar energy received by the Earth is: 

&lt;math&gt;\pi \times \text {radius}_{earth}^2 \times 1366 \frac{W}{m^2} &lt;/math&gt; 

Plugging in the values this gives:

 3.14159 * 6,378,100 * 6,378,100 * 1,366 = 174,575,322,671,459,743 watts 

or rounding off this number it is: 1.74 * 10&lt;sup&gt;17&lt;/sup&gt; watts or 170,000 TWh where a terrawatt is 1 * 10&lt;sup&gt;12&lt;/sup&gt; watts

Clearly this value is not exact because the Earth is in an elliptic orbit and so the amount of solar energy varies slightly throughout the year and also the Earth bulges ever so slightly. But these effects will not change the overall magnitude of the figure. 

What matters though is the amount of solar energy hitting the ground at any point and this varies depending on the latitude, the season, the local time and the thickness of the atmosphere that it has to travel through. In general at least 30% of the 1.36 kWh is lost in the atmosphere while at higher latitudes where the sunlight has to pass through more atmosphere because it is entering at an angle it can probably be as high as 50%. The effect of season is most pronounced at higher latitudes.

A point to note about local time, is that mid-day local time is generally defined as the time at which the sun reaches the highest point in the sky at that location. In reality then there is just a few hours in the day when the sun is around it's maximum angular height in the sky and supplying most energy

==The Vision==
It has long been the dream of many solar proponents that solar energy can provide all our energy needs. In 2004, the '''worldwide energy consumption''' of the human race was on average 15 TW (= 1.5 x 10&lt;sup&gt;13&lt;/sup&gt; W).&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=xqV4Gs4hKoEC&amp;newbks=0&amp;printsec=frontcover&amp;pg=PA9&amp;dq=worldwide+energy+consumption+&amp;hl=en|title=Energy Efficiency and Human Activity: Past Trends, Future Prospects|last=Schipper|first=Lee|last2=Meyers|first2=Stephen|date=1992-11-19|publisher=Cambridge University Press|isbn=978-0-521-43297-9|language=en}}&lt;/ref&gt; The figures above show that the Earth receives more than 10,000 times this amount of energy and show clearly there is enough to provide all our energy needs.

It is hard to argue with these figures and it has been suggested many times that the one off gift of nature of fossil fuel energy should be used to build the solar energy infrastructure which can last humankind indefinitely given that other problems like ecological collapse, global warming and a pollution crisis are avoided.

A popular calculation that has been done is to determine how many square kilometers would need to be covered by solar energy to capture enough energy for our needs. The figure widely quoted is a box about 100 miles (or 160 km) per side which is 10,000 square miles (25,600 km&lt;sup&gt;2&lt;/sup&gt;) but this figure is just for the United States.

The calculation is derived roughly as follows. It is assumed that the location is at a relatively low latitude in sunny place like a desert and that 1000 w/m&lt;sup&gt;2&lt;/sup&gt; are received at ground level. Taking current solar photovoltaic cell efficiencies actually deployed, we assume a value of around 10% efficiency. Therefore to achieve 15 Twh (10&lt;sup&gt;12&lt;/sup&gt;w):

  Area = 15 Twh / 1000 w/m&lt;sup&gt;2&lt;/sup&gt; * 0.1 = 1.5 * 10&lt;sup&gt;11&lt;/sup&gt; m&lt;sup&gt;2&lt;/sup&gt;

Taking the square root and converting from m&lt;sup&gt;2&lt;/sup&gt; to km&lt;sup&gt;2&lt;/sup&gt; gives:

387,298 metres or '''387 km''' per side. Doing the calculation using an efficiency of 15% instead of 10% gives a value of '''316 km''' (~197 miles) which is considerably lower. And an even higher efficiency of 20% which would be less likely for such a large system would reduce this to '''274 km'''.

If this calculation was done for just the total global amount of electricity used then it would be lower. Other factors to consider are that by introducing more efficient energy technologies and greatly increasing the amount of public transport would greatly reduce the amount of oil and thereby total energy usage.

==The Needs==

=Technologoies=
[[Solar Energy Technologies|Solar Energy Technologies]]

==Solar Electric==
The generation of electricity from solar energy is dominated by the use of solar photovoltaic cells. However there is the less well known solar thermovoltaic effect which uses heat instead to convert solar infrared light into electricity directly.
===Solar [[Photovoltaics]]===
These convert sunlight into directly electricity. For a full discussion of this process see [[[w:Solar_Cell]]].

There are three main types of solar cells. These are '''crystalline silicon solar cells, amphorous silicon solar cells''' and '''thin film solar cells'''.

====Crystalline Solar Cells====
These were the first types of solar cells and require a very pure silicon crystal to be grown from which the solar cell is manufactured. The disadvanages are that the crystals are time consuming and require a lot of energy to grow them. This is because the silicon is held at a high temperature. Indeed this is one of the reasons why it has been so difficult to rapidly increase global production of crystalline solar cells.

As a result of this, the energy payback time of crystalline solar cells has been typically 10 to 20 years, although this is being reduced slowly as the technologies improves and the cells are being manufactured using thin slices of silicon.

One of the main advantages though of crystalline cells is that they tend to have the highest efficiencies of all solars and over the past decade or so, figures from 25% to just above 30% have been achieved. Typically values in the field are less.

====Amorphous Solar Cells====

====Thin Film Solar Cells====
Thin Film solar cells cover a diverse range of cells types using a number of different chemical elements. For a full discussion of this see [[w:Solar_Cell#Thin_films]].

The main advantages of thin film cells is that their energy payback times tend to be lower and they tend to be much more flexible and less brittle. Some types can be even bent and rolled up as sheets.

Thin film cells currently cover about 10% or less of total global production of solar cells.

One of the less discussed problems with thin film solar cells is that they tend to use both relatively rare and toxic chemicals which could present longer term environmental problems.

====Concentrating Systems====
Silicon crystalline solar cells can be used in conjunction with solar concentrates to increase the amount of sunlight following on a given cell. Some types of solar cells even achieve slightly higher efficiencies by doing this.

The main advantages of employing this technique is that you can use much more expensive and higher effeciency cells and at the same time use collect a bigger area of sunlight using cheaper components and thereby reducing your overall cost.

===Solar Thermovoltaics===
These type of solar cells operate by converting infrared light into electricity. They are effectively the same as solar photovoltaics except they are just operating at a longer wavelength. The sun produces a significant amount of infrared radiation as to any hot objects like furnaces. In theory these cells can also be used to operate off the infrared emitted by furnaces.

===Solar Tower===
These are still largely in the experimental phase. They are often referred to as solar updraft tower because they consist of a canopy over a relatively large area under which the area is heated. The canopy is angled upward towards the base of the tower and this allows the heated air to flow under convention up the tower. Significant updraft is created and this can be enough to drive a turbine in the tower shaft.

===Solar Stirling Engine===

==Solar Heating and Cooling==
In addition to transmitting energy in the visible spectrum, the Sun transmits energy in the infrared spectrum. Because materials such as water and metal are good at absorbing wavelengths in this spectrum, solar thermal presents an opportunity to reduce water and space heating needs by supplementing them with solar energy. 

Solar Heating can be further classified into active and passive systems. Examples of active systems are solar water panels that you often see on roofs for heating water, whereas a south facing building that takes advantage of the sun can be considered as a passive form of solar energy.

Solar heating and cooling is the less glamorous side of solar energy compared to the more high tech solar photovoltaics. However it is an important area because significant amounts of energy are used for heating water and for heating and cooling buildings. In the USA, electrical demand is actually higher in the summer rather than the winter because of the amount of electrical powered air-conditioning use. There are considerable opportunities for replacing and reducing the amount of fossil fuel energy to supply these services.

===Solar Panels===
Solar panels are now used quite extensively domestically especially in Mediterranean countries and Israel for heating water.

They basically consist of a radiator painted black to absorb the sunlight and covered by glass to trap the heat. The water is circulated through the narrow piping so that it has a relatively long distance to travel and therefore gives it time to pick up the heat.

At present there is a huge range of panel designs and manufacturers. Solar panels are the ultimate for the do it yourself (DIY) enthusiast because anyone with basic plumbing skills should be able to put together a relatively simple system. Vacuum tube seems to offer the best solution as a reflector is placed under them and this enables 360 Degree sunlight around the tube effectively using the whole tube while the sun shines. Temperature of 303 Degree Celsius has been measured of the inner copper tip which is used on high pressure systems. The low pressure system has no inner copper tube however temperature of 75 degree has been measured. The time is now to make use of natural elements and eliminate destroying the planet. Lets use this technology, its free.

===Solar Chimney===
A solar chimney is a way of improving the natural ventilation of buildings by using convection of air heated by the sun.

See [[w:Solar_chimney]]

==Passive Solar Energy==

===Passive Solar Heating and Cooling===

====Building Orientation====

===Passive Solar Cooling===

====Thick Walls====

===Passive Solar Lighting===

====Sky Lights====

====Solar Tubes====

=Generation of Electricity=
[[Image:Dye sensitized cell phys.jpg|300px|thumb|Dye sensitized solar cell]]
[[Generation of Electricity by Solar|Generation of Electricity by Solar]]

==Utility Factor==
The utility factor refers to the fraction of time in a day that can be used to generate solar power.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=udxvAAAAQBAJ&amp;newbks=0&amp;printsec=frontcover&amp;q=The+utility+factor+refers+to+the+fraction+of+time+in+a+day+that+can+be+used+to+generate+solar+power.&amp;hl=en|title=Solar Energy: Technologies and Project Delivery for Buildings|last=Walker|first=Andy|date=2013-08-07|publisher=John Wiley &amp; Sons|isbn=978-1-118-41654-9|language=en}}&lt;/ref&gt; Clearly average over a year, it is going to be a theoretical maximum of only 50%. In practice this value will be even lower because if you account for the period around both sunrise and sunset when the sun is at a very low angle in the sky, there will not be sufficient power from the available light to generate any significant amounts of power. Taking this into consideration then would suggest a value of 40% being more typical.

It might be worth noting that the typical utility factors achieved for wind power so far are around 26% on average with the best sites as high as 35%.

Utility factors for fossil fuel power stations burning coal, gas, oil and nuclear power stations can be as high as 80% to 90% since other than maintenance there are no other limitations. In situations of a severe drought though, the availability of fresh water for cooling can impact this.

Therefore to replace a given electricial capacity of a fossil fuel power station typically requires double the capacity when being replaced by a solar electrical plant. So for example to replace a 500 MW coal plant would require an installation of 1000 MW (1GW) of solar.

===Capacity and Energy Usage===
When discussing any energy supply it is important to always distinguish between figures for installed capacity and actual energy produced. One should look out for units of wattage which are usually quoted for capacity and kilowatt hours (Kwh) which are used for power produced.

In a traditional fossil fuel power station, it typically might run at close to full power at its utility factor. Its yearly power output will then be approximately:

   365 days * 24 hours * 0.8 Utility Factor * Power Station Capacity (MW)

'''For a 500 MW power station this is:  365 * 24 * 0.8 * 500 = 3,504,000 MWh =''' 3.5 TwH

A figure worth remembering is that there are approximately '''8,760 hours per year'''.

==Residential Power==
The generation of electricity by solar photovoltaic for residential use has been growing reasonably steadily over the years. It is still at a very small level, certainly far less than 1% of domestic electrical consumption. 

For a residential setting the quantity of power needed varies but it ranges from about 2Kw to 4Kw systems. They can be setup as '''grid connected''' or '''off-grid'''. For off-grid systems a battery bank is needed for storing power and this can add to the cost.

==Commercial Power==

==Utility Scale Power==

==Open educational resources==

* [https://ecampusontario.pressbooks.pub/solarpv/ Solar Photovoltaics for Design Engineers]

===Wikipedia===
* [[w:Dye-sensitized solar cells|Dye-sensitized solar cells]]
* [[w:List of photovoltaics companies|List of photovoltaics companies]]
* [[w:Agrivoltaics|Agrivoltaics]]

== Tables of solar energy equations ==
{| class=&quot;wikitable&quot;
|-
! scope=&quot;col&quot; width=&quot;200&quot; | Standard solar test conditions (common name/s) 
! scope=&quot;col&quot; width=&quot;125&quot; | (Common) symbol/s 
! scope=&quot;col&quot; width=&quot;125&quot; | SI unit
! scope=&quot;col&quot; width=&quot;100&quot; | Dimension
|-
! Standard solar irradiance
| ''G''
| 1
| kWm&lt;sup&gt;−2&lt;/sup&gt;
|-
! Standard
| ''Tcell''
| 25
| °C
|-
|}

=== Solar thermal equations ===
Useful thermal heat output:

&lt;math&gt;\dot Q''_u = \frac {\dot m_f \cdot C_{p,f} \cdot (T_{out} - T_{in})}{A}
&lt;/math&gt;

Instantaneous thermal efficiency of a solar thermal collector:

&lt;math&gt;\eta_c = \frac {\dot Q''_u}{\dot G''}&lt;/math&gt;

==See also==
[[Renewable energy/Solar thermal]]

== External links ==
* Youtube: [http://www.youtube.com/watch?v=qYeynLy6pj8 How solar panels are assembled]
* [http://www.appropedia.org/Category:Photovoltaics Appropedia:Category:Photovoltaics]
* September 2008 ''[http://www.nss.org/news/releases/pr20080909.html First-of-a-Kind Long-Distance Demonstration of Solar-Powered Wireless Power Transmission Technology]''
* June 2008 ''[http://www.eurekalert.org/pub_releases/2008-06/epfd-neb062708.php New efficiency benchmark for dye-sensitized solar cells]''
* October 2008 ''[http://www.makeitrightnola.org/ Brad Pitt's green building project in New Orlean's Lower 9th Ward]''

[[Category:Renewable energy]]

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      <text bytes="17047" sha1="npff7jp0rbnzv6x2rn95pcb7q1qiies" xml:space="preserve">[[Image:Solar concentrator.jpg|thumb|right|300px|[[Topic:Solar energy|Solar energy]] can be concentrated using a parabolic reflector to achieve very high temperatures.]]

Welcome to the topic of solar energy! In this topic area you will find courses devoted to the engineering and science of solar energy, and its diverse applications and systems. Please see the page on [[renewable energy]] for other renewable energy applications.

Solar energy systems come in a diverse range of technologies and are largely divided up into two main categories which are solar electric and solar heating. Solar electric applications covers solar photovoltaic cells used for generating electricity, while solar heating systems typically are used for heating water.

A further subdivision could be thought of as '''active''' and '''passive''' systems. Active systems are those which are built solely to capture solar energy whereas as passive systems are more generally design features that make use of the available solar energy. An example of the latter is building architecture.

== Courses ==
[[Introduction to solar energy]]

==The Resource==
[[Solar energy/Resource availability]]

At the top of the atmosphere the solar irradiance on the Earth is 1,366 watts per square meter (w/m-&lt;sup&gt;2&lt;/sup&gt;).&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=y_GMTXRtxJ8C&amp;newbks=0&amp;printsec=frontcover&amp;pg=PA36&amp;dq=At+the+top+of+the+atmosphere+the+solar+irradiance+on+the+Earth+is+1,366+watts+per+square+meter+(w/m-2).&amp;hl=en|title=Climate Change: Myths and Realities|publisher=Jeevananda Reddy|language=en}}&lt;/ref&gt; This means that the total solar energy received by the Earth is: 

&lt;math&gt;\pi \times \text {radius}_{earth}^2 \times 1366 \frac{W}{m^2} &lt;/math&gt; 

Plugging in the values this gives:

 3.14159 * 6,378,100 * 6,378,100 * 1,366 = 174,575,322,671,459,743 watts 

or rounding off this number it is: 1.74 * 10&lt;sup&gt;17&lt;/sup&gt; watts or 170,000 TWh where a terrawatt is 1 * 10&lt;sup&gt;12&lt;/sup&gt; watts

Clearly this value is not exact because the Earth is in an elliptic orbit and so the amount of solar energy varies slightly throughout the year and also the Earth bulges ever so slightly. But these effects will not change the overall magnitude of the figure. 

What matters though is the amount of solar energy hitting the ground at any point and this varies depending on the latitude, the season, the local time and the thickness of the atmosphere that it has to travel through. In general at least 30% of the 1.36 kWh is lost in the atmosphere while at higher latitudes where the sunlight has to pass through more atmosphere because it is entering at an angle it can probably be as high as 50%. The effect of season is most pronounced at higher latitudes.

A point to note about local time, is that mid-day local time is generally defined as the time at which the sun reaches the highest point in the sky at that location. In reality then there is just a few hours in the day when the sun is around it's maximum angular height in the sky and supplying most energy

==The Vision==
It has long been the dream of many solar proponents that solar energy can provide all our energy needs. In 2004, the '''worldwide energy consumption''' of the human race was on average 15 TW (= 1.5 x 10&lt;sup&gt;13&lt;/sup&gt; W).&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=xqV4Gs4hKoEC&amp;newbks=0&amp;printsec=frontcover&amp;pg=PA9&amp;dq=worldwide+energy+consumption+&amp;hl=en|title=Energy Efficiency and Human Activity: Past Trends, Future Prospects|last=Schipper|first=Lee|last2=Meyers|first2=Stephen|date=1992-11-19|publisher=Cambridge University Press|isbn=978-0-521-43297-9|language=en}}&lt;/ref&gt; The figures above show that the Earth receives more than 10,000 times this amount of energy and show clearly there is enough to provide all our energy needs.

It is hard to argue with these figures and it has been suggested many times that the one off gift of nature of fossil fuel energy should be used to build the solar energy infrastructure which can last humankind indefinitely given that other problems like ecological collapse, global warming and a pollution crisis are avoided.

A popular calculation that has been done is to determine how many square kilometers would need to be covered by solar energy to capture enough energy for our needs. The figure widely quoted is a box about 100 miles (or 160 km) per side which is 10,000 square miles (25,600 km&lt;sup&gt;2&lt;/sup&gt;) but this figure is just for the United States.

The calculation is derived roughly as follows. It is assumed that the location is at a relatively low latitude in sunny place like a desert and that 1000 w/m&lt;sup&gt;2&lt;/sup&gt; are received at ground level. Taking current solar photovoltaic cell efficiencies actually deployed, we assume a value of around 10% efficiency. Therefore to achieve 15 Twh (10&lt;sup&gt;12&lt;/sup&gt;w):

  Area = 15 Twh / 1000 w/m&lt;sup&gt;2&lt;/sup&gt; * 0.1 = 1.5 * 10&lt;sup&gt;11&lt;/sup&gt; m&lt;sup&gt;2&lt;/sup&gt;

Taking the square root and converting from m&lt;sup&gt;2&lt;/sup&gt; to km&lt;sup&gt;2&lt;/sup&gt; gives:

387,298 metres or '''387 km''' per side. Doing the calculation using an efficiency of 15% instead of 10% gives a value of '''316 km''' (~197 miles) which is considerably lower. And an even higher efficiency of 20% which would be less likely for such a large system would reduce this to '''274 km'''.

If this calculation was done for just the total global amount of electricity used then it would be lower. Other factors to consider are that by introducing more efficient energy technologies and greatly increasing the amount of public transport would greatly reduce the amount of oil and thereby total energy usage.

==The Needs==

=Technologoies=
[[Solar Energy Technologies|Solar Energy Technologies]]

==Solar Electric==
The generation of electricity from solar energy is dominated by the use of solar photovoltaic cells. However there is the less well known solar thermovoltaic effect which uses heat instead to convert solar infrared light into electricity directly.
===Solar [[Photovoltaics]]===
These convert sunlight into directly electricity. For a full discussion of this process see [[[w:Solar_Cell]]].

There are three main types of solar cells. These are '''crystalline silicon solar cells, amphorous silicon solar cells''' and '''thin film solar cells'''.

====Crystalline Solar Cells====
These were the first types of solar cells and require a very pure silicon crystal to be grown from which the solar cell is manufactured. The disadvanages are that the crystals are time consuming and require a lot of energy to grow them. This is because the silicon is held at a high temperature. Indeed this is one of the reasons why it has been so difficult to rapidly increase global production of crystalline solar cells.

As a result of this, the energy payback time of crystalline solar cells has been typically 10 to 20 years, although this is being reduced slowly as the technologies improves and the cells are being manufactured using thin slices of silicon.

One of the main advantages though of crystalline cells is that they tend to have the highest efficiencies of all solars and over the past decade or so, figures from 25% to just above 30% have been achieved. Typically values in the field are less.

====Amorphous Solar Cells====

====Thin Film Solar Cells====
Thin Film solar cells cover a diverse range of cells types using a number of different chemical elements. For a full discussion of this see [[w:Solar_Cell#Thin_films]].

The main advantages of thin film cells is that their energy payback times tend to be lower and they tend to be much more flexible and less brittle. Some types can be even bent and rolled up as sheets.

Thin film cells currently cover about 10% or less of total global production of solar cells.

One of the less discussed problems with thin film solar cells is that they tend to use both relatively rare and toxic chemicals which could present longer term environmental problems.

====Concentrating Systems====
Silicon crystalline solar cells can be used in conjunction with solar concentrates to increase the amount of sunlight following on a given cell. Some types of solar cells even achieve slightly higher efficiencies by doing this.

The main advantages of employing this technique is that you can use much more expensive and higher effeciency cells and at the same time use collect a bigger area of sunlight using cheaper components and thereby reducing your overall cost.

===Solar Thermovoltaics===
These type of solar cells operate by converting infrared light into electricity. They are effectively the same as solar photovoltaics except they are just operating at a longer wavelength. The sun produces a significant amount of infrared radiation as to any hot objects like furnaces. In theory these cells can also be used to operate off the infrared emitted by furnaces.

===Solar Tower===
These are still largely in the experimental phase. They are often referred to as solar updraft tower because they consist of a canopy over a relatively large area under which the area is heated. The canopy is angled upward towards the base of the tower and this allows the heated air to flow under convention up the tower. Significant updraft is created and this can be enough to drive a turbine in the tower shaft.

===Solar Stirling Engine===

==Solar Heating and Cooling==
In addition to transmitting energy in the visible spectrum, the Sun transmits energy in the infrared spectrum. Because materials such as water and metal are good at absorbing wavelengths in this spectrum, solar thermal presents an opportunity to reduce water and space heating needs by supplementing them with solar energy. 

Solar Heating can be further classified into active and passive systems. Examples of active systems are solar water panels that you often see on roofs for heating water, whereas a south facing building that takes advantage of the sun can be considered as a passive form of solar energy.

Solar heating and cooling is the less glamorous side of solar energy compared to the more high tech solar photovoltaics. However it is an important area because significant amounts of energy are used for heating water and for heating and cooling buildings. In the USA, electrical demand is actually higher in the summer rather than the winter because of the amount of electrical powered air-conditioning use. There are considerable opportunities for replacing and reducing the amount of fossil fuel energy to supply these services.

===Solar Panels===
Solar panels are now used quite extensively domestically especially in Mediterranean countries and Israel for heating water.

They basically consist of a radiator painted black to absorb the sunlight and covered by glass to trap the heat. The water is circulated through the narrow piping so that it has a relatively long distance to travel and therefore gives it time to pick up the heat.

At present there is a huge range of panel designs and manufacturers. Solar panels are the ultimate for the do it yourself (DIY) enthusiast because anyone with basic plumbing skills should be able to put together a relatively simple system. Vacuum tube seems to offer the best solution as a reflector is placed under them and this enables 360 Degree sunlight around the tube effectively using the whole tube while the sun shines. Temperature of 303 Degree Celsius has been measured of the inner copper tip which is used on high pressure systems. The low pressure system has no inner copper tube however temperature of 75 degree has been measured. The time is now to make use of natural elements and eliminate destroying the planet. Lets use this technology, its free.

===Solar Chimney===
A solar chimney is a way of improving the natural ventilation of buildings by using convection of air heated by the sun. As cooler air is heated up by flow adjacent to a relatively warmer surface, the air warms. As air obeys the ideal gas law, as temperature increases, density decreases, leading to a buoyancy induced flow upwards through the chimney.

See [[w:Solar_chimney]]

==Passive Solar Energy==

===Passive Solar Heating and Cooling===

====Building Orientation====

===Passive Solar Cooling===

====Thick Walls====

===Passive Solar Lighting===

====Sky Lights====

====Solar Tubes====

=Generation of Electricity=
[[Image:Dye sensitized cell phys.jpg|300px|thumb|Dye sensitized solar cell]]
[[Generation of Electricity by Solar|Generation of Electricity by Solar]]

==Utility Factor==
The utility factor refers to the fraction of time in a day that can be used to generate solar power.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=udxvAAAAQBAJ&amp;newbks=0&amp;printsec=frontcover&amp;q=The+utility+factor+refers+to+the+fraction+of+time+in+a+day+that+can+be+used+to+generate+solar+power.&amp;hl=en|title=Solar Energy: Technologies and Project Delivery for Buildings|last=Walker|first=Andy|date=2013-08-07|publisher=John Wiley &amp; Sons|isbn=978-1-118-41654-9|language=en}}&lt;/ref&gt; Clearly average over a year, it is going to be a theoretical maximum of only 50%. In practice this value will be even lower because if you account for the period around both sunrise and sunset when the sun is at a very low angle in the sky, there will not be sufficient power from the available light to generate any significant amounts of power. Taking this into consideration then would suggest a value of 40% being more typical.

It might be worth noting that the typical utility factors achieved for wind power so far are around 26% on average with the best sites as high as 35%.

Utility factors for fossil fuel power stations burning coal, gas, oil and nuclear power stations can be as high as 80% to 90% since other than maintenance there are no other limitations. In situations of a severe drought though, the availability of fresh water for cooling can impact this.

Therefore to replace a given electricial capacity of a fossil fuel power station typically requires double the capacity when being replaced by a solar electrical plant. So for example to replace a 500 MW coal plant would require an installation of 1000 MW (1GW) of solar.

===Capacity and Energy Usage===
When discussing any energy supply it is important to always distinguish between figures for installed capacity and actual energy produced. One should look out for units of wattage which are usually quoted for capacity and kilowatt hours (Kwh) which are used for power produced.

In a traditional fossil fuel power station, it typically might run at close to full power at its utility factor. Its yearly power output will then be approximately:

   365 days * 24 hours * 0.8 Utility Factor * Power Station Capacity (MW)

'''For a 500 MW power station this is:  365 * 24 * 0.8 * 500 = 3,504,000 MWh =''' 3.5 TwH

A figure worth remembering is that there are approximately '''8,760 hours per year'''.

==Residential Power==
The generation of electricity by solar photovoltaic for residential use has been growing reasonably steadily over the years. It is still at a very small level, certainly far less than 1% of domestic electrical consumption. 

For a residential setting the quantity of power needed varies but it ranges from about 2Kw to 4Kw systems. They can be setup as '''grid connected''' or '''off-grid'''. For off-grid systems a battery bank is needed for storing power and this can add to the cost.

==Commercial Power==

==Utility Scale Power==

==Open educational resources==

* [https://ecampusontario.pressbooks.pub/solarpv/ Solar Photovoltaics for Design Engineers]

===Wikipedia===
* [[w:Dye-sensitized solar cells|Dye-sensitized solar cells]]
* [[w:List of photovoltaics companies|List of photovoltaics companies]]
* [[w:Agrivoltaics|Agrivoltaics]]

== Tables of solar energy equations ==
{| class=&quot;wikitable&quot;
|-
! scope=&quot;col&quot; width=&quot;200&quot; | Standard solar test conditions (common name/s) 
! scope=&quot;col&quot; width=&quot;125&quot; | (Common) symbol/s 
! scope=&quot;col&quot; width=&quot;125&quot; | SI unit
! scope=&quot;col&quot; width=&quot;100&quot; | Dimension
|-
! Standard solar irradiance
| ''G''
| 1
| kWm&lt;sup&gt;−2&lt;/sup&gt;
|-
! Standard
| ''Tcell''
| 25
| °C
|-
|}

=== Solar thermal equations ===
Useful thermal heat output:

&lt;math&gt;\dot Q''_u = \frac {\dot m_f \cdot C_{p,f} \cdot (T_{out} - T_{in})}{A}
&lt;/math&gt;

Instantaneous thermal efficiency of a solar thermal collector:

&lt;math&gt;\eta_c = \frac {\dot Q''_u}{\dot G''}&lt;/math&gt;

==See also==
[[Renewable energy/Solar thermal]]

== External links ==
* Youtube: [http://www.youtube.com/watch?v=qYeynLy6pj8 How solar panels are assembled]
* [http://www.appropedia.org/Category:Photovoltaics Appropedia:Category:Photovoltaics]
* September 2008 ''[http://www.nss.org/news/releases/pr20080909.html First-of-a-Kind Long-Distance Demonstration of Solar-Powered Wireless Power Transmission Technology]''
* June 2008 ''[http://www.eurekalert.org/pub_releases/2008-06/epfd-neb062708.php New efficiency benchmark for dye-sensitized solar cells]''
* October 2008 ''[http://www.makeitrightnola.org/ Brad Pitt's green building project in New Orlean's Lower 9th Ward]''

[[Category:Renewable energy]]

{{BookCat}}</text>
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  <page>
    <title>Climatology</title>
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      <text bytes="1678" sha1="3jx706ulidusr79h2mukqcfqy51jk3z" xml:space="preserve">{{environmental science}}
Climatology is a branch of atmospheric science as well as geographical and environmental science.

==Overview==
Climate is a dynamical system that is based upon the long term averages and behaviour of Earth's atmosphere both on the global scale and the regional scale. This topic aims to help your understanding of the basics of climate science as well as the many impactors climate has on the Earth as well as the impactors the Earth has on the climate.

== Subpages ==
{{Subpages/List}}

==History==
* A brief history of Climatology

== Outline ==

===Level 1: The Earths' Radiation Budget===

====Circulations====
* Atmospheric Circulation
* Oceanic Circulation

====Cycles====
* Diurnal Cycle
* Seasonal Cycle

====Perturbations====
* Short term (1 year to 1,000 years)
* Middle term (1,000 years to 10,000 years)
* Long term (10,000 years to 1,000,000 years)

----
(Original)
* Weather forecasting
* Weather phenomena
* Climate basics
* Weather observations and other data
* Solar forcing
* Atmospheric Circulation
* Ocean circulation
* Earth radiation budget
* Glaciation and Sea Ice cover
* Atmospheric physics
* Climate models, General Circulation Models

== Related learning materials ==
* Climate Change
* Global warming
* The Ozone Hole
* [[Carbon_capture_and_storage|Carbon capture and storage]]

===Related disciplines===
* [[Paleoclimatology]]
* [[Weather satellites]]
* [[Remote sensing]]
* [[Ecological engineering]]

===External learning resources===
* [http://www.realclimate.org/ RealClimate.org]

== See also ==
* [[Aviation Weather]]
* [[School:Geography]]
* [[School:Meteorology]]


[[Category:Meteorology]]
[[Category:Geography]]</text>
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{{TOCright}}

== Classical guitar Methods ==

*[[Gaspar Sanz]] ([[1640]]-[[1710]]) - ''[[Instucción de música sobre la Guitarra Española]]'' 
*[[Federico Moretti]] - ''Principios para tocar la guitarra de seis òrdenes'' (Madrid, 1799)
*[[Ferdinando Carulli]] (1770-1841) (Italy) - ''Méthode complette'', op.27 (Paris, 1810)
*[[Fernando Sor]] (1778-1839) (Spain)

&quot;Method for the Spanish Guitar&quot;

First published in French under the name “[[Méthode pour la Guitare]]” (Paris, [[1830]]) and then translated in English under the name “Method for the Spanish Guitar” in 1832. This Famous method for the guitar has justly been called &quot;the most remarkable book ever published on the guitar&quot;.

*[[Dionisio Aguado]] (1784-1849)
&quot;Guitar School&quot; (Escuela de guitarra, Madrid, 1825).

''Nuevo Méthode para Guitarra'' (Madrid, 1843)

*[[Matteo Carcassi]] (1792-1853) (Italy) - ''Méthode complète pour la guitare'', op.59 (Paris, 1836)

*[[Francisco Tárrega]]  died before he was able to conclude his intention to publish a method of guitar at the request of the violinist Juan Manen.

*[[Emilio Pujol]] (1886-1980) (Spain) - [http://www.lmd.jussieu.fr/~polcher/pujol_tot/pujol.html The Emilio Pujol guitar method], by Daniela Polcher. (French)

* [[Abel Carlevaro]] (1918-2002) School of the Guitar

*[[Suzuki method]]
Published in seven volumes.

*[[Kodály Method]]
[http://www.egtaguitarforum.org/ExtraArticles/kodaly.html] By Luke Dunlea

Additional incomplete information is available here: [[Classical guitar methods]]

==Famous guitar teachers==

===Nineteenth century===

*[[Fernando Sor]] (1778-1839) (Spain)
**Teachers: 
**Students: [[Napoléon Coste]] ([[1806]]-[[1883]]) ([[France]])
*[[Dionisio Aguado]] ([[1784]]-[[1849]]) ([[Spain]])
**Teachers: 
**Students:
*[[Francisco Tárrega]] (1852—1909) (Spain)
**Teachers: [[Julián Arcas]]
**Students: [[Emilio Pujol]] (1886-1980) (Spain), [[Miguel Llobet]] (1878-1938)

===Twentieth century===

*[[Miguel Llobet]] ([[1878]]-[[1938]]) 
**Teachers: [[Francisco Tárrega]]
**Students:[[María Luisa Anido]]

*[[Emilio Pujol]] (1886-1980) (Spain) 
**Teachers: [[Francisco Tárrega]]
**Students: [[Alberto Ponce]] (*1935)

*[[Regino Sainz de la Maza]]
**Teachers: 
**Students: [[Alirio Diaz]] (*[[1923]]) 

*[[Andrés Segovia]] ([[1893]]-[[1987]]) 
**Teachers: Self-taught, but influenced by [[Miguel Llobet]]
**Students: [[John Williams (guitarist)|John Williams]] (*1941) ([[Australia]]), *[[Eliot Fisk]] (*1958), [[José Tomás]]

*[[Aaron Shearer]]
**Teachers:
**Students: [[Manuel Barrueco]]

*[[José Tomás]] ([[1934]]-[[2001]])
**Teachers: [[Andrés Segovia]], [[Emilio Pujol]] 
**Students: 

*[[Alberto Ponce]] (*[[1935]])
**Teachers: [[Emilio Pujol]]
**Students: [[Roland Dyens]] (*[[1955]]) ([[France]])

*[[Abel Carlevaro]] ([[1916]]-[[2001]])
**Teachers: [[Andrés Segovia]]
**Students: [[Eduardo Fernandez]], [[Marcelo Kayath]]

===Classical guitar technique===

{{Main|classical guitar technique}}

===Bibliography===

[http://guitarra.artelinkado.com/guitarra/cejilla_julio_gimeno.htm Bibliography at the end of the article]

===External links===

* [http://hobbies.expertvillage.com/interviews/classical-guitar.htm Classical guitar pedagogy on expertvillage.com]


[[Category:Guitar]]</text>
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    <title>Protein Science I</title>
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This is an introductory course in the science of [[w:protein|proteins]], aimed at undergraduate biochemistry majors at a research university.  It will cover the composition, structure and function of proteins, as well as experimental methods for producing, purifying and studying proteins.  Basic topics in chemistry, such as molecular bonds and interactions, will be touched upon as well.

This course falls within the [[Topic:Biochemistry|Department of Biochemistry]] within the [[School:Biology|School of Biology]].

==Syllabus==

===Week 1: Molecules and molecular forces: putting the chemistry back into biochemistry===

* Daily lesson 1.1: The periodic table
* Daily lesson 1.2: Electronegativity
* Daily lesson 1.3: Chemical (covalent) bonds and ionic bonds
* Daily lesson 1.4: Resonance stabilization
* Daily lesson 1.5: Chemical reactions, 
* Daily lesson 1.6: Chemical reaction kinetics and free energy
* Daily lesson 1.7: pH and pK&lt;sub&gt;a&lt;/sub&gt;

===Week 2: Non-covalent molecular forces===

* Daily lesson 2.1: Titration curves
* Daily lesson 2.2: pK&lt;sub&gt;a&lt;/sub&gt; shifts in molecules
* Daily lesson 2.3: Charge, electrostatics, screening
* Daily lesson 2.4: van der Waals interactions
* Daily lesson 2.5: Hydrogen bonds
* Daily lesson 2.6: Hydrophobic interactions
* Daily lesson 2.7: Cation-pi and aromatic-aromatic interactions

===Week 3: Amino acids, peptides, and proteins===

* Daily lesson 3.1: Overview of amino acids and their metabolism
* Daily lesson 3.2: Hydrophobic amino acids (VILMA, FYW, CPG)
* Daily lesson 3.3: Polar and charged amino acids (STNQ, DERKH)
* Daily lesson 3.4: Peptide bond
* Daily lesson 3.5: Peptides
* Daily lesson 3.6: Proteins
* Daily lesson 3.7: Disulfide bonds

===Week 4: Protein primary structure===

* Daily lesson 4.1: Protein sequencing: Edman degradation and mass spectrometry
* Daily lesson 4.2: Post-translational modifications
* Daily lesson 4.3:
* Daily lesson 4.4:
* Daily lesson 4.5: Alignment of two sequences: BLAST, Needleman-Wunsch, Smith-Waterman
* Daily lesson 4.6: Multiple sequence alignment and molecular evolution
* Daily lesson 4.7: Multiple sequence alignment and homology modeling

===Week 5: Protein secondary structure===

* Daily lesson 5.1: Definition of a dihedral angle
* Daily lesson 5.2: Dihedral angles found in proteins
* Daily lesson 5.3: Sidechain rotamers
* Daily lesson 5.4: Backbone dihedrals: the Ramachandran map
* Daily lesson 5.5: Alpha helices
* Daily lesson 5.6: Beta sheets and turns
* Daily lesson 5.7: Conformational stability of secondary structure in solution

===Week 6: ===

* Daily lesson 6.1:
* Daily lesson 6.2:
* Daily lesson 6.3:
* Daily lesson 6.4:
* Daily lesson 6.5:
* Daily lesson 6.6:
* Daily lesson 6.7:

===Week 7: ===

* Daily lesson 7.1:
* Daily lesson 7.2:
* Daily lesson 7.3:
* Daily lesson 7.4:
* Daily lesson 7.5:
* Daily lesson 7.6:
* Daily lesson 7.7:

===Week 8: ===

* Daily lesson 8.1:
* Daily lesson 8.2:
* Daily lesson 8.3:
* Daily lesson 8.4:
* Daily lesson 8.5:
* Daily lesson 8.6:
* Daily lesson 8.7:

===Week 9: ===

* Daily lesson 9.1:
* Daily lesson 9.2:
* Daily lesson 9.3:
* Daily lesson 9.4:
* Daily lesson 9.5:
* Daily lesson 9.6:
* Daily lesson 9.7:

===Week 10: ===

* Daily lesson 10.1:
* Daily lesson 10.2:
* Daily lesson 10.3:
* Daily lesson 10.4:
* Daily lesson 10.5:
* Daily lesson 10.6:
* Daily lesson 10.7:

==Instructors==

* [[User:Proteins|Prof. Bill Wedemeyer]], from [[w:Michigan State University|Michigan State University]]

[[Category:Biochemistry]]</text>
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    <title>Quantitative biology I</title>
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This is an introductory course in the mathematical and computational methods useful to students of the life sciences.  The course is not intended to recapitulate a full degree program in physics, mathematics or computer science.  Rather, the goal is to provide hands-on training, so that the student can ''use'' already available techniques and canned software, even if they are not ready to develop new tools.  The course is intended to be useful to students in a wide variety of disciplines, e.g., biochemistry, genetics, cellular biology, microbiology, physiology, systems biology and metabolomics.

This course falls within the [[Topic:Biochemistry|Department of Biochemistry]] within the [[School:Biology|School of Biology]].

==Rough syllabus==

The plan is to present the topics in the following order; the rationale is that each topic builds on those preceding it.

* Basic math
* Basic statistics
* Statistical correlations
* Curve fitting
* Noise and noise reduction
* Diffusion, estimates of hydrodynamic size and molecular weight
* Matrix methods, computational efficiency
* Kinetics, linear and nonlinear
* Equilibrium; thermodynamics and statistical mechanics
* Electronic structure, spectroscopy and covalent bonds
* Non-covalent molecular forces
* Nuclear isotopes and NMR
* Resolution in various guises
* Fourier transforms
* Normal modes, molecular dynamics, Monte Carlo methods
* Computational predictions in molecular biology

Alternatively, these topics can be sorted into four major groups

* Group 1: Statistical topics
**Basic statistics, statistical correlations, curve fitting
**Noise and noise reduction
**Diffusion and hydrodynamic methods

* Group 2: Matrix methods and kinetics
** Matrix methods, computational efficiency
** Kinetics, linear and nonlinear
** Equilibrium; thermodynamics and statistical mechanics

* Group 3: Chemical/physical topics
** Electronic structure, spectroscopy, and covalent bonds
** Non-covalent molecular forces
** Nuclear isotopes and NMR

* Group 4: Mathematical and computational methods
** Fourier transforms and linear differential equations
** The concept of resolution
** X-ray crystallography
** Molecular dynamics, Monte Carlo methods, normal modes

It is anticipated that the full curriculum would correspond to two semester-long courses.

==Detailed syllabus==

This is an initial draft; the timings should not be taken seriously.

===Week 1: Basic mathematics===

* Daily lesson 1.1: Units; basic measurements: gel-box volume
* Daily lesson 1.2: Solution math:serial dilutions, colony-forming units
* Daily lesson 1.3: Calibration of a pipette
* Daily lesson 1.4: pH, pI; maleic/fumaric acids
* Daily lesson 1.5: linear, semilog and log-log plots; polar plots
* Daily lesson 1.6: asymptotic behavior; perturbations
* Daily lesson 1.7: powers of 10; Fermi problems

===Week 2: Basic statistics===

* Daily lesson 2.1: discrete vs. continuous stochastic variables
* Daily lesson 2.2: 1-dimensional statistical distributions; error bars vs. confidence limits
* Daily lesson 2.3: 
* Daily lesson 2.4: 
* Daily lesson 2.5: 
* Daily lesson 2.6: 
* Daily lesson 2.7: 

===Week 3: ===

* Daily lesson 3.1: 
* Daily lesson 3.2: 
* Daily lesson 3.3: 
* Daily lesson 3.4: 
* Daily lesson 3.5: 
* Daily lesson 3.6: 
* Daily lesson 3.7: 

===Week 4: ===

* Daily lesson 4.1: 
* Daily lesson 4.2: 
* Daily lesson 4.3:
* Daily lesson 4.4:
* Daily lesson 4.5: 
* Daily lesson 4.6: 
* Daily lesson 4.7: 

===Week 5: ===

* Daily lesson 5.1: 
* Daily lesson 5.2: 
* Daily lesson 5.3: 
* Daily lesson 5.4: 
* Daily lesson 5.5: 
* Daily lesson 5.6: 
* Daily lesson 5.7: 
===Week 6: ===

* Daily lesson 6.1:
* Daily lesson 6.2:
* Daily lesson 6.3:
* Daily lesson 6.4:
* Daily lesson 6.5:
* Daily lesson 6.6:
* Daily lesson 6.7:

===Week 7: ===

* Daily lesson 7.1:
* Daily lesson 7.2:
* Daily lesson 7.3:
* Daily lesson 7.4:
* Daily lesson 7.5:
* Daily lesson 7.6:
* Daily lesson 7.7:

===Week 8: ===

* Daily lesson 8.1:
* Daily lesson 8.2:
* Daily lesson 8.3:
* Daily lesson 8.4:
* Daily lesson 8.5:
* Daily lesson 8.6:
* Daily lesson 8.7:

===Week 9: ===

* Daily lesson 9.1:
* Daily lesson 9.2:
* Daily lesson 9.3:
* Daily lesson 9.4:
* Daily lesson 9.5:
* Daily lesson 9.6:
* Daily lesson 9.7:

===Week 10: ===

* Daily lesson 10.1:
* Daily lesson 10.2:
* Daily lesson 10.3:
* Daily lesson 10.4:
* Daily lesson 10.5:
* Daily lesson 10.6:
* Daily lesson 10.7:

==Instructors==

* [[User:Proteins|Prof. Bill Wedemeyer]], from [[w:Michigan State University|Michigan State University]]

[[Category:Biochemistry]]
[[Category:Life sciences]]</text>
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  <page>
    <title>Correlation</title>
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'''Correlation''' (co-relation) refers to the degree of relationship (or dependency) between two variables.

Linear correlation refers to straight-line relationships between two [[variable]]s. 

A correlation can range between -1 (perfect negative relationship) and +1 (perfect positive relationship), with 0 indicating no straight-line relationship.

The earliest known use of correlation was in the late 19th century[http://jeff560.tripod.com/c.html].

==Introduction==
[[Image:Venn-diagram-AB.png|thumb|right|275px|The degree of linear relationship between two variables can be represented in terms of a [[w:Venn Diagram|Venn Diagram]]. Perfectly overlapping circles would indicate a correlation of 1, and non-overlapping circles would represent a correlation of 0.]]
When we ask questions such as &quot;Is X related to Y?&quot;, &quot;Does X predict Y?&quot;, and &quot;Does X account for Y&quot;?, we are interested in measuring and better understanding the relationship between two variables.

Correlation measures the extent to which variables:
# covary
# depend on one another
# predict one another

The extent of correlation between two variables, by convention, is denoted ''r'', and the correlation between variable X and variable Y is indicated by ''r''&lt;sub&gt;XY&lt;/sub&gt;.

Correlations are standardised to vary between -1 and +1, with 0 representing no relationship, -1 a perfect negative relationship, and +1 a perfect positive relationship.

A variety of bivariate correlational statistics are available, the choice of which depends on the variables' [[Levels of measurement|level of measurement]]:
*Nominal by nominal: [[Contingency table]], [[Pearson's chi-square test]], [[Phi]]/[[Cramer's V]]
*Ordinal by ordinal: [[Spearman's rho]], [[Kendall's tau-b]]
*Dichotomous by interval/ratio: [[Point biserial correlation coefficient]]
*Interval/ration by interval/ratio: [[Pearson product-moment correlation coefficient]]

Correlational analyses should be accompanied by appropriate bivariate graphs, such as:
*Nominal by nominal: [[Clustered bar chart]]s
*Ordinal by ordinal: [[Scatterplot]] (with point bins)
*Interval/ratio by interval/ratio: [[Scatterplot]]

==The world is made of covariation==
[[File:Bee 1 by andy205.jpg|thumb|right|250px|Bees and flowers tend to co-occur.]]
Responses which vary can be measured as a variable (i.e., responses are distributed across a range).

Responses to two or more variables may covary. These variables share some variation. When the value of one variable is high, the value of other variable tends to be high (positive correlation) or low (negative correlation).

If you look around, you may notice that the world is made of covariation! e.g.,
* pollen count is positively correlated with bee activity
* rainfall is positively correlated with amount of vegetation
* hours of study is positively correlated with test performance
* number of fire trucks attending a fire is correlated with cost of repairs for the fire[http://www.burns.com/wcbspurcorl.htm]
* Sibling's IQ is positively correlated
* perceived air temperature is negatively correlated with amount of clothing worn

The more you look, the more you'll see that there are many predictable patterns of co-occurrence between  phenomena (i.e., things tend to occur together).

==Scatterplots==
Independent variable (IV) (predictor) is placed on the X axis and dependent variable (DV) is placed on the Y axis. Each case is plotted according to its X and Y value.
[[File:Scatterplot r=-.76.png|thumb|center|400px|''r'' = .76]]
==Visual inspection of scatterplots is essential==
It is unwise to rely solely on correlation as a statistic that indicates the nature of the relationship between variables without also examining a visualisation of the data such as through a scatterplot.

For example, the linear (straight-line) correlation in each of these four scatterplots is .82, yet the nature of what the data indicated about the relationship between the variables is very different for each.

[[Image:Anscombe's quartet 3.svg|thumb|400px|center|Four sets of data with the same correlation of 0.816]]
{{expand section}}

==Homoscedasticity==
[[File:Homoscedasticity.png|thumb|right|200px|Scatterplot showing homoscedasticity.]]
[[File:Heteroscedasticity.png|thumb|right|200px|Scatterplot showing heteroscedasticity.]]
If the data are normally distributed, then scatterplots should be homoscedastistic (even spread about the line of best fit).

If data are not normally distributed (e.g., skewed), then the bivariate distribution may be heteroscedastic (uneven spread about the line of best fit). This violate the assumption of homoscedasticity for correlation.

For more information, see [[w:Homoscedasticity|Homoscedasticity]] and [[w:Heteroscedasticity|Heteroscedasticity]] (Wikipedia).
{{clear}}
==Correlation does not equal causation==
Correlation does not prove causation, although it may be consistent with causation. It is important to understand that correlation does not equal causation. A relationship between two variables may be caused by a third variable.
[[File:Correlation vs causation.png|center|400px|thumb|[http://www.burns.com/wcbspurcorl.htm Spurious correlations]. [[w:Correlation does not imply causation|Correlation does not imply causation]] (Wikipedia). [http://answers.google.com/answers/threadview?id=368317%22 More examples] ]] 
{{expand section}}
See [[w:Correlation does not imply causation|Correlation does imply causation]] (Wikipedia)

==Range restriction==
[[Image:correlation range dependence.svg|400px|center|thumb|[[w:Pearson product moment correlation coefficient|Pearson]]/[[w:Spearman's rank correlation coefficient|Spearman]] correlation coefficients between ''X'' and ''Y'' are shown when the two variables' ranges are unrestricted, and when the range of ''X'' is restricted to the interval (0,1).]]
{{expand section}}

For more info, see [https://books.google.com.au/books?id=5WFohzuwzP0C&amp;pg=PA281&amp;lpg=PA281#v=onepage&amp;q&amp;f=false the effect of range restrictions] (Howell, 2009) and [http://davidmlane.com/hyperstat/A68809.html restricted range] (Lane, n. d.).

For a practical tutorial, see [[Survey research and design in psychology/Tutorials/Correlation/Outliers and restricted range|outliers and restricted range]].

==Coefficient of determination==
When a correlation coefficient (''r'') is squared (''r''&lt;sup&gt;2&lt;/sup&gt;), this gives the '''coefficient of determination''' which is the percentage of variance shared between the two variables.

;See also
* [http://www.slideshare.net/jtneill/linear-correlation/63 Lecture slide]
* References
** Allen &amp; Bennett, 2010, p. 173 
** Howell, 2010, p. 344
* [[w:Coefficient of determination|Coefficient of determination]] (Wikipedia)

==Interactive activity==
'''Correlation guess''': [[Survey research and design in psychology/Tutorials/Correlation/Correlation guess|Correlation guess]]

==Quiz==
'''Test yourself''': This is a pre-quiz to see what you already know - [[/Introductory quiz|Introductory quiz]]

==See also==
* [[Survey research and design in psychology/Lectures/Correlation|Correlation]] (Lecture)
* [[Survey research and design in psychology/Tutorials/Correlation|Correlation]] (Tutorial)
* [[w:Correlation and dependence|Correlation]] (Wikipedia)

==External links==
* [http://www.mat.ufrgs.br/~viali/estatistica/mat2282/material/textos/Eleven.pdf 11 ways to look at the chi-squared coefficient for contingency tables]
* [http://data.psych.udel.edu/laurenceau/PSYC861Regression/READINGS/rodgers-nicewander-1988-r-13-ways.pdf 13 ways to look at the correlation coefficient]
* [http://www.statisticalengineering.com/correlation.htm Correlation] (Annis, 2008)
* [http://www2.chass.ncsu.edu/garson/pa765/correl.htm Correlation] (Garson, 2008)
* [http://www.uwsp.edu/psych/stat/7/correlat.htm Correlation] (Plonsky, 2006)
* [http://www.socialresearchmethods.net/kb/statcorr.php Correlation] (Trochim, 2006)
* [http://www.sportsci.org/resource/stats/correl.html Correlation coefficient] (Hopkins, 2000)
* [http://www.andrews.edu/~calkins/math/edrm611/edrm05.htm Correlation coefficients] (Calkins, 2005)
* [http://obereed.net/hh/correlation.html New poll shows correlation is causation] (Humour)
* [http://faculty.vassar.edu/lowry/ch3b.html Rank order correlation] (Lowry, 2008)
* [http://ucspace.canberra.edu.au/display/7126/Tutorial+-+Linear+correlation Tutorial - Correlation] (Neill, 2010)
* [http://www.mega.nu/ampp/rummel/uc.htm Understanding correlation] (Rummel, 1976)

[[Category:Linear correlation| ]]</text>
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      <text bytes="2235" sha1="mx0i3af7sydcmfb0m5hwe0pheu82lvl" xml:space="preserve">{{TOCright}}
==Thesis==
My PhD [[thesis]] has been written in the field of [[educational psychology]] and, more specifically, on the psycho-educational effectiveness of [[outdoor education]] programs. It examines changes in [[life effectiveness]] self-ratings by ~3000 outdoor education participants, most of whom were involved in [[w:Outward Bound|Outward Bound]] Australia programs from 1992 to 2000, and finds, overall moderately positive and lasting impacts on enhancement of life skills, with little notable variation, although longer, challenging programs with motivated adults were approximately twice as effective as shorter, compulsory programs with school-aged students.

The main contributions of the thesis were:
* Major reviews of theoretical aspects of outdoor education and research about the effects of outdoor education.
* Further development and testing of the Life Effectiveness Questionnaire.
* Largest outcome study of outdoor education programs to date.
* Demonstration of use of effect sizes for analysis of change.

The main limitations of this thesis included that:
* The sample was heavily reliant on programs conducted by one organisation (Outward Bound Australia) during the 1990s.
* The study did not measure any antecedent or process-type variables (including individual differences) which may have helped to account for variability in change.
* The study was quantitative and thus lacked somewhat in qualitative insight into the nature of OE program's personal and social development effects.

==Status==
* Submitted for examination to the [[University of Western Sydney]], June, 2008
* Examiner feedback received, August, 2008
** Examiner 1 recommended &quot;Accept&quot;
** Examiner 2 recommended &quot;Minor changes&quot;
* Submitted final draft with examiner-requested changes - ~18 October, 2008
&lt;!-- * [http://wilderdom.com/phd More info] (including draft) --&gt;

==Planned publications==
{{center top}}
 
{| border=1 cellspacing=0 cellpadding=5 style =&quot;background:transparent;&quot;
|-
! Title/Topic
! Draft due
|-
| Psychometrics of the LEQ
| December 2008
|-
| Effects of OE programs
| February 2008
|- 
| Review of research
| July 2008
|-
| Theoretical model review
| September 2008
|}
{{center bottom}}

[[Category:PhD]]</text>
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==Reference==
{{Hanging indent|1=Neill, J. T. (2008). ''Enhancing life effectiveness: The impacts of outdoor education programs''. [Doctoral dissertation, Western Sydney University]. https://researchers.westernsydney.edu.au/en/studentTheses/enhancing-life-effectiveness-the-impacts-of-outdoor-education-pro/}}

==Overview==
My PhD [[thesis]] has been written in the field of [[educational psychology]] and, more specifically, on the psycho-educational effectiveness of [[outdoor education]] programs. It examines changes in [[life effectiveness]] self-ratings by ~3000 outdoor education participants, most of whom were involved in [[w:Outward Bound|Outward Bound]] Australia programs from 1992 to 2000, and finds, overall moderately positive and lasting impacts on enhancement of life skills, with little notable variation, although longer, challenging programs with motivated adults were approximately twice as effective as shorter, compulsory programs with school-aged students.

The main contributions of the thesis were:
* Major reviews of theoretical aspects of outdoor education and research about the effects of outdoor education.
* Further development and testing of the Life Effectiveness Questionnaire.
* Largest outcome study of outdoor education programs to date.
* Demonstration of use of effect sizes for analysis of change.

The main limitations of this thesis included that:
* The sample was heavily reliant on programs conducted by one organisation (Outward Bound Australia) during the 1990s.
* The study did not measure any antecedent or process-type variables (including individual differences) which may have helped to account for variability in change.
* The study was quantitative and thus lacked somewhat in qualitative insight into the nature of OE program's personal and social development effects.

==Status==
* Submitted for examination to the [[University of Western Sydney]], June, 2008
* Examiner feedback received, August, 2008
** Examiner 1 recommended &quot;Accept&quot;
** Examiner 2 recommended &quot;Minor changes&quot;
* Submitted final draft with examiner-requested changes - ~18 October, 2008
&lt;!-- * [http://wilderdom.com/phd More info] (including draft) --&gt;

==Planned publications==
{{center top}}
 
{| border=1 cellspacing=0 cellpadding=5 style =&quot;background:transparent;&quot;
|-
! Title/Topic
! Draft due
|-
| Psychometrics of the LEQ
| December 2008
|-
| Effects of OE programs
| February 2008
|- 
| Review of research
| July 2008
|-
| Theoretical model review
| September 2008
|}
{{center bottom}}

[[Category:PhD]]</text>
      <sha1>imw452c378p9lijz5rfx5r8ah79ek4y</sha1>
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==Reference==
{{Hanging indent|1=Neill, J. T. (2008). ''Enhancing life effectiveness: The impacts of outdoor education programs''. [Doctoral dissertation, University of Western Sydney]. https://researchers.westernsydney.edu.au/en/studentTheses/enhancing-life-effectiveness-the-impacts-of-outdoor-education-pro/}}

==Overview==
My PhD [[thesis]] has been written in the field of [[educational psychology]] and, more specifically, on the psycho-educational effectiveness of [[outdoor education]] programs. It examines changes in [[life effectiveness]] self-ratings by ~3000 outdoor education participants, most of whom were involved in [[w:Outward Bound|Outward Bound]] Australia programs from 1992 to 2000, and finds, overall moderately positive and lasting impacts on enhancement of life skills, with little notable variation, although longer, challenging programs with motivated adults were approximately twice as effective as shorter, compulsory programs with school-aged students.

The main contributions of the thesis were:
* Major reviews of theoretical aspects of outdoor education and research about the effects of outdoor education.
* Further development and testing of the Life Effectiveness Questionnaire.
* Largest outcome study of outdoor education programs to date.
* Demonstration of use of effect sizes for analysis of change.

The main limitations of this thesis included that:
* The sample was heavily reliant on programs conducted by one organisation (Outward Bound Australia) during the 1990s.
* The study did not measure any antecedent or process-type variables (including individual differences) which may have helped to account for variability in change.
* The study was quantitative and thus lacked somewhat in qualitative insight into the nature of OE program's personal and social development effects.

==Status==
* Submitted for examination to the [[University of Western Sydney]], June, 2008
* Examiner feedback received, August, 2008
** Examiner 1 recommended &quot;Accept&quot;
** Examiner 2 recommended &quot;Minor changes&quot;
* Submitted final draft with examiner-requested changes - ~18 October, 2008
&lt;!-- * [http://wilderdom.com/phd More info] (including draft) --&gt;

==Planned publications==
{{center top}}
 
{| border=1 cellspacing=0 cellpadding=5 style =&quot;background:transparent;&quot;
|-
! Title/Topic
! Draft due
|-
| Psychometrics of the LEQ
| December 2008
|-
| Effects of OE programs
| February 2008
|- 
| Review of research
| July 2008
|-
| Theoretical model review
| September 2008
|}
{{center bottom}}

[[Category:PhD]]</text>
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==[[../Research/Profiles|Profiles]]==
{{../Research/Profiles}}
&lt;!-- ==2026==
--&gt;
==2025==
{{Hanging indent|1=
Brichacek, A., Neill, J. T., Murray, K., Rieger, E., Watsford, C. (2025). Body Image Flexibility and Inflexibility Scale (BIFIS). In W. Ramseyer Winter, T. L. Tylka, &amp; A. M. Landor (Eds.), ''Handbook of body image-related measures''. Cambridge University Press (pp. 118–121). https://doi.org/10.1017/9781009398275.039

{{User:Jtneill/Publications/2025/Body}}&lt;!--
Neill, J. T., Herbert, S., Hartley, R., &amp; D'Cunha, N. (in preparation). ''Art for Wellbeing at the National Gallery of Australia: Thematic analysis of participant and staff perspectives''.

Lozancic Babic, V. &amp; Neill, J. T. ...  --&gt;
}}

==2024==
{{Hanging indent|1=

Black, H. M., &amp; Neill, J. T. (2024). Wellbeing through nature: A qualitative exploration of psychosocial aspects of a Landcare ACT nature-connection program. ''Journal of Outdoor and Environmental Education''. https://doi.org/10.1007/s42322-024-00184-2

Boerma, M., Beel, N., Neill, J. T., Jeffries, C., Krishnamoorthy, G., &amp; Guerri-Guttenberg, J. (2024). Male-friendly counselling for young men: a thematic analysis of client and caregiver experiences of Menslink counselling. ''Australian Psychologist'', 1–12. https://doi.org/10.1080/00050067.2024.2378119 ([https://www.researchgate.net/publication/385649063_Male-friendly_counselling_for_young_men_a_thematic_analysis_of_client_and_caregiver_experiences_of_Menslink_counselling#fullTextFileContent pdf])

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (accepted). The Body Image Flexibility and Inflexibility Scale (BIFIS). In V. Ramseyer Winter, T.  Tylka, &amp; A. Landor (Eds.), ''Handbook of body image-related measures'' (pp. *–*). Cambridge University Press.

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2024). The distinct affect regulation functions of body image flexibility and inflexibility: A prospective study in adolescents and emerging adults. ''Body Image'', ''50'', 101726. https://doi.org/10.1016/j.bodyim.2024.101726

{{User:Jtneill/Publications/2024/Collaborative}}

Neill, J. T. &amp; Black. H. (2024). ''[https://landcareact.org.au/wp-content/uploads/2026/01/Wellbeing-through-Nature-Final-Report.pdf Landcare ACT Wellbeing through Nature program evaluation: Final report]''. University of Canberra, Australia.

{{User:Jtneill/Publications/2024/Rich}}

}}

==2023==
{{Hanging indent|1=
Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2023). Ways of responding to body image threats: Development of the Body Image Flexibility and Inflexibility Scale for Youth. ''Journal of Contextual Behavioral Science'', ''30'', 31–40. https://doi.org/10.1016/j.jcbs.2023.08.007

{{/2023/WIL}}

Ross, B. M., &amp; Neill, J. T. (2023). Exploring the relationship between mental health, drug use, personality, and attitudes towards psilocybin-assisted therapy. ''[https://akjournals.com/view/journals/2054/2054-overview.xml Journal of Psychedelic Studies]'', ''7''(2), 114–118. https://doi.org/10.1556/2054.2023.00264}}

==2022==
{{Hanging indent|1=

Neill, J. T., Goch, I., Sullivan, A., &amp; Simons, M. (2022). The role of burn camp in the recovery of young people from burn injury: A qualitative study using long-term follow-up interviews with parents and participants. ''Burns'', ''48''(5), 1139–1148. https://doi.org/10.1016/j.burns.2021.09.020

Stevenson, D. J., Neill, J. T., Ball, K., Smith, R., &amp; Shores, M. C. (2022). How do preschool to year 6 educators prevent and cope with occupational violence from students? ''Australian Journal of Education'', ''66''(2), 154–170. https://doi.org/10.1177/00049441221092472. [https://www.teachermagazine.com/au_en/articles/the-research-files-episode-77-coping-with-violence-from-students Podcast].

}}

==2021==
{{Hanging indent|1=Brichacek, A. L., Murray, K., Neill, J. T., &amp; Rieger, E. (2021). Contextual behavioral approaches to understanding body image threats and coping in youth: A qualitative study. ''Journal of Adolescent Research'', ''39''(2), 328–360. https://doi.org/10.1177/07435584211007851}}

==2020==
{{Hanging indent|1=Boerma, M., &amp; Neill, J. (2020). The role of grit and self-control in university student academic achievement and satisfaction. ''College Student Journal'', ''54''(4), 431–442.

Boerma, M., Neill, J., &amp; Brown, P. (2020). Perseverance of effort moderates the relationship between psychological distress and life satisfaction. ''European Journal of Applied Positive Psychology'', ''4''(16), 1–11. https://www.nationalwellbeingservice.org/volumes/volume-4-2020/volume-4-article-16/}}

==2018==
{{Hanging indent|1=Neill, J. T. (2018). ''[https://menslink.org.au/wp-content/uploads/2018/10/UC-Report-into-Long-term-Impacts-of-Menslink-Counselling-and-Mentoring-Oct-2018.pdf Long-term impacts of Menslink counselling and mentoring]''. University of Canberra.}}

==2017==
{{Hanging indent|1=Booth, J. W., &amp; Neill, J. T. (2017). Coping strategies and the development of psychological resilience. ''Journal of Outdoor and Environmental Education'', ''20''(1), 47–54. https://doi.org/10.1007/BF03401002}}

==2016==
{{Hanging indent|1=

Bowen, D. J., &amp; Neill, J. T. (2016). Effects of the PCYC Catalyst outdoor adventure intervention program on youths' life skills, mental health, and delinquent behaviour. ''International Journal of Adolescence and Youth'', ''21''(1), 34–55. https://doi.org/10.1080/02673843.2015.1027716

Bowen, D. J., Neill, J. T., &amp; Crisp, S. J. (2016). Wilderness adventure therapy effects on the mental health of youth participants. ''Evaluation and Program Planning'', ''58'', 49–59. https://doi.org/10.1016/j.evalprogplan.2016.05.005

{{/2016/Internationalisation}}}}

==2013==
{{Hanging indent|1=
Bowen, D. J., &amp; Neill, J. T. (2013). A meta-analysis of adventure therapy outcomes and moderators. ''The Open Psychology Journal'', ''6''(1). http://dx.doi.org/10.2174/1874350120130802001
{{/2013/Promoting}}

{{/2013/Teaching}}
}}

==2011==
{{Hanging indent|1=

Gray, T. L. &amp; Neill, J. T. (2011). Program evaluation. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 164–182). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.010

Neill, J. T., &amp; Gray, T. L. (2011). Technology, risk and outdoor programming. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 132–149). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.008
}}

==2010==
Mackay, G. J., &amp; Neill, J. T. (2010). The effect of “green exercise” on state anxiety and the role of exercise duration, intensity, and greenness: A quasi-experimental study. ''Psychology of Sport and Exercise'', ''11''(3), 238–245. https://doi.org/10.1016/j.psychsport.2010.01.002

==2008==
{{Hanging indent|1=Neill, J. T. (2008). Enhancing life effectiveness: The impacts of outdoor education programs. [Unpublished doctoral dissertation]. University of Western Sydney. https://researchers.westernsydney.edu.au/en/studentTheses/enhancing-life-effectiveness-the-impacts-of-outdoor-education-pro/}}

==2005==
{{Hanging indent|1=Fabrizio, S. M., &amp; Neill, J. T. (2005). Cultural adaptation in outdoor programming. ''Journal of Outdoor and Environmental Education'', ''9''(2), 44–56. https://doi.org/10.1007/BF03400820}}

==2002==
{{/2002/Dramaturgy}}

==1997==
{{Hanging indent|1=
Hattie, J., Marsh, H. W., Neill, J. T., &amp; Richards, G. E. (1997). Adventure education and Outward Bound: Out-of-class experiences that make a lasting difference. ''Review of Educational Research'', ''67''(1), 43–87. https://doi.org/10.3102/00346543067001043}}

==Reports==
{{Hanging indent|1=
Neill, J. T. &amp; Bowen, D. J. (2014). ''[https://drive.google.com/file/d/0B2N4zSp4hmN9WUF3bzhuZ3JoNGM/view?usp=sharing&amp;resourcekey=0-y0ZTjcdhHXqQKNtz50BW0A Research evaluation of PCYC Bornhoffen Catalyst intervention programs for youth-at-risk &lt;nowiki&gt;[&lt;/nowiki&gt;2012-2013&lt;nowiki&gt;]&lt;/nowiki&gt;]''. University of Canberra.
}}

==Theses==
* [[User:Jtneill/PhD|PhD]]
&lt;!-- 
==Published==
* [http://www.wilderdom.com/JamesNeill/JamesNeillpublications.htm Articles &amp; presentations by James Neill]
--&gt;

==Ideas / In progress==
* [[User:Jtneill/4 pillars of free and open teaching|4 pillars of free and open teaching]]
* Some international trends in outdoor education - Past, present, and future
* Ingando camp (life effectiveness)
* Life Effectiveness Questionnaire psychometrics
* OE outcomes (longitudinal study)
* Adolescent Coping Scale psychometrics
* Resilience Scale psychometrics
* Overview of Outdoor Education Theory and/or Research
* Overview of Outdoor Education in Australia
* Overview of Adventure Therapy Theory and/or Education
* Past Trends and Future Directions for Outdoor Education
* Psychological Aspects of Outdoor Education
* Outdoor Education and Modern Technology
* Outdoor Education and Environmental Sustainability

==See also==
* [[User:Jtneill/Presentations]]</text>
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==[[../Research/Profiles|Profiles]]==
{{../Research/Profiles}}
&lt;!-- ==2026==
--&gt;
==2025==
{{Hanging indent|1=
Brichacek, A., Neill, J. T., Murray, K., Rieger, E., Watsford, C. (2025). Body Image Flexibility and Inflexibility Scale (BIFIS). In W. Ramseyer Winter, T. L. Tylka, &amp; A. M. Landor (Eds.), ''Handbook of body image-related measures''. Cambridge University Press (pp. 118–121). https://doi.org/10.1017/9781009398275.039

{{User:Jtneill/Publications/2025/Body}}&lt;!--
Neill, J. T., Herbert, S., Hartley, R., &amp; D'Cunha, N. (in preparation). ''Art for Wellbeing at the National Gallery of Australia: Thematic analysis of participant and staff perspectives''.

Lozancic Babic, V. &amp; Neill, J. T. ...  --&gt;
}}

==2024==
{{Hanging indent|1=

Black, H. M., &amp; Neill, J. T. (2024). Wellbeing through nature: A qualitative exploration of psychosocial aspects of a Landcare ACT nature-connection program. ''Journal of Outdoor and Environmental Education''. https://doi.org/10.1007/s42322-024-00184-2

Boerma, M., Beel, N., Neill, J. T., Jeffries, C., Krishnamoorthy, G., &amp; Guerri-Guttenberg, J. (2024). Male-friendly counselling for young men: a thematic analysis of client and caregiver experiences of Menslink counselling. ''Australian Psychologist'', 1–12. https://doi.org/10.1080/00050067.2024.2378119 ([https://www.researchgate.net/publication/385649063_Male-friendly_counselling_for_young_men_a_thematic_analysis_of_client_and_caregiver_experiences_of_Menslink_counselling#fullTextFileContent pdf])

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (accepted). The Body Image Flexibility and Inflexibility Scale (BIFIS). In V. Ramseyer Winter, T.  Tylka, &amp; A. Landor (Eds.), ''Handbook of body image-related measures'' (pp. *–*). Cambridge University Press.

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2024). The distinct affect regulation functions of body image flexibility and inflexibility: A prospective study in adolescents and emerging adults. ''Body Image'', ''50'', 101726. https://doi.org/10.1016/j.bodyim.2024.101726

{{User:Jtneill/Publications/2024/Collaborative}}

Neill, J. T. &amp; Black. H. (2024). ''[https://landcareact.org.au/wp-content/uploads/2026/01/Wellbeing-through-Nature-Final-Report.pdf Landcare ACT Wellbeing through Nature program evaluation: Final report]''. University of Canberra, Australia.

{{User:Jtneill/Publications/2024/Rich}}

}}

==2023==
{{Hanging indent|1=
Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2023). Ways of responding to body image threats: Development of the Body Image Flexibility and Inflexibility Scale for Youth. ''Journal of Contextual Behavioral Science'', ''30'', 31–40. https://doi.org/10.1016/j.jcbs.2023.08.007

{{/2023/WIL}}

Ross, B. M., &amp; Neill, J. T. (2023). Exploring the relationship between mental health, drug use, personality, and attitudes towards psilocybin-assisted therapy. ''[https://akjournals.com/view/journals/2054/2054-overview.xml Journal of Psychedelic Studies]'', ''7''(2), 114–118. https://doi.org/10.1556/2054.2023.00264}}

==2022==
{{Hanging indent|1=

Neill, J. T., Goch, I., Sullivan, A., &amp; Simons, M. (2022). The role of burn camp in the recovery of young people from burn injury: A qualitative study using long-term follow-up interviews with parents and participants. ''Burns'', ''48''(5), 1139–1148. https://doi.org/10.1016/j.burns.2021.09.020

Stevenson, D. J., Neill, J. T., Ball, K., Smith, R., &amp; Shores, M. C. (2022). How do preschool to year 6 educators prevent and cope with occupational violence from students? ''Australian Journal of Education'', ''66''(2), 154–170. https://doi.org/10.1177/00049441221092472. [https://www.teachermagazine.com/au_en/articles/the-research-files-episode-77-coping-with-violence-from-students Podcast].

}}

==2021==
{{Hanging indent|1=Brichacek, A. L., Murray, K., Neill, J. T., &amp; Rieger, E. (2021). Contextual behavioral approaches to understanding body image threats and coping in youth: A qualitative study. ''Journal of Adolescent Research'', ''39''(2), 328–360. https://doi.org/10.1177/07435584211007851}}

==2020==
{{Hanging indent|1=Boerma, M., &amp; Neill, J. (2020). The role of grit and self-control in university student academic achievement and satisfaction. ''College Student Journal'', ''54''(4), 431–442.

Boerma, M., Neill, J., &amp; Brown, P. (2020). Perseverance of effort moderates the relationship between psychological distress and life satisfaction. ''European Journal of Applied Positive Psychology'', ''4''(16), 1–11. https://www.nationalwellbeingservice.org/volumes/volume-4-2020/volume-4-article-16/}}

==2018==
{{Hanging indent|1=Neill, J. T. (2018). ''[https://menslink.org.au/wp-content/uploads/2018/10/UC-Report-into-Long-term-Impacts-of-Menslink-Counselling-and-Mentoring-Oct-2018.pdf Long-term impacts of Menslink counselling and mentoring]''. University of Canberra.}}

==2017==
{{Hanging indent|1=Booth, J. W., &amp; Neill, J. T. (2017). Coping strategies and the development of psychological resilience. ''Journal of Outdoor and Environmental Education'', ''20''(1), 47–54. https://doi.org/10.1007/BF03401002}}

==2016==
{{Hanging indent|1=

Bowen, D. J., &amp; Neill, J. T. (2016). Effects of the PCYC Catalyst outdoor adventure intervention program on youths' life skills, mental health, and delinquent behaviour. ''International Journal of Adolescence and Youth'', ''21''(1), 34–55. https://doi.org/10.1080/02673843.2015.1027716

Bowen, D. J., Neill, J. T., &amp; Crisp, S. J. (2016). Wilderness adventure therapy effects on the mental health of youth participants. ''Evaluation and Program Planning'', ''58'', 49–59. https://doi.org/10.1016/j.evalprogplan.2016.05.005

{{/2016/Internationalisation}}}}

==2013==
{{Hanging indent|1=
Bowen, D. J., &amp; Neill, J. T. (2013). A meta-analysis of adventure therapy outcomes and moderators. ''The Open Psychology Journal'', ''6''(1). http://dx.doi.org/10.2174/1874350120130802001
{{/2013/Promoting}}

{{/2013/Teaching}}
}}

==2011==
{{Hanging indent|1=

Gray, T. L. &amp; Neill, J. T. (2011). Program evaluation. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 164–182). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.010

Neill, J. T., &amp; Gray, T. L. (2011). Technology, risk and outdoor programming. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 132–149). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.008
}}

==2010==
Mackay, G. J., &amp; Neill, J. T. (2010). The effect of “green exercise” on state anxiety and the role of exercise duration, intensity, and greenness: A quasi-experimental study. ''Psychology of Sport and Exercise'', ''11''(3), 238–245. https://doi.org/10.1016/j.psychsport.2010.01.002

==2008==
{{Hanging indent|1=Neill, J. T. (2008). Enhancing life effectiveness: The impacts of outdoor education programs. [Doctoral dissertation, University of Western Sydney]. University of Western Sydney. https://researchers.westernsydney.edu.au/en/studentTheses/enhancing-life-effectiveness-the-impacts-of-outdoor-education-pro/}}

==2005==
{{Hanging indent|1=Fabrizio, S. M., &amp; Neill, J. T. (2005). Cultural adaptation in outdoor programming. ''Journal of Outdoor and Environmental Education'', ''9''(2), 44–56. https://doi.org/10.1007/BF03400820}}

==2002==
{{/2002/Dramaturgy}}

==1997==
{{Hanging indent|1=
Hattie, J., Marsh, H. W., Neill, J. T., &amp; Richards, G. E. (1997). Adventure education and Outward Bound: Out-of-class experiences that make a lasting difference. ''Review of Educational Research'', ''67''(1), 43–87. https://doi.org/10.3102/00346543067001043}}

==Reports==
{{Hanging indent|1=
Neill, J. T. &amp; Bowen, D. J. (2014). ''[https://drive.google.com/file/d/0B2N4zSp4hmN9WUF3bzhuZ3JoNGM/view?usp=sharing&amp;resourcekey=0-y0ZTjcdhHXqQKNtz50BW0A Research evaluation of PCYC Bornhoffen Catalyst intervention programs for youth-at-risk &lt;nowiki&gt;[&lt;/nowiki&gt;2012-2013&lt;nowiki&gt;]&lt;/nowiki&gt;]''. University of Canberra.
}}

==Theses==
* [[User:Jtneill/PhD|PhD]]
&lt;!-- 
==Published==
* [http://www.wilderdom.com/JamesNeill/JamesNeillpublications.htm Articles &amp; presentations by James Neill]
--&gt;

==Ideas / In progress==
* [[User:Jtneill/4 pillars of free and open teaching|4 pillars of free and open teaching]]
* Some international trends in outdoor education - Past, present, and future
* Ingando camp (life effectiveness)
* Life Effectiveness Questionnaire psychometrics
* OE outcomes (longitudinal study)
* Adolescent Coping Scale psychometrics
* Resilience Scale psychometrics
* Overview of Outdoor Education Theory and/or Research
* Overview of Outdoor Education in Australia
* Overview of Adventure Therapy Theory and/or Education
* Past Trends and Future Directions for Outdoor Education
* Psychological Aspects of Outdoor Education
* Outdoor Education and Modern Technology
* Outdoor Education and Environmental Sustainability

==See also==
* [[User:Jtneill/Presentations]]</text>
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'''Life effectiveness''' is term which refers to the performance of generic personal and social life skills. Related terms include [[psychological resilience]] and [[practical intelligence]].

The '''Life Effectiveness Questionnaire''' provides a self-report instrument for assessing several purported life effectiveness skills, including:
# '''Time Management'''
# '''Social Competence'''
# '''Achievement Motivation'''
# '''Intellectual Flexibility'''
# '''Emotional Control'''
# '''Task Leadership'''
# '''Active Initiative'''
# '''Self Confidence'''

==Workshop activity==
# Present and discuss notion of life effectiveness - handout - for more info, see http://wilderdom.com/leq
## Related concepts: Practical intelligence, self-concept etc.
## Definitions of generic life effectiveness skills
## Describe the eight life effectiveness (LEQ-H) dimensions
# Present and discuss outdoor education results from http://wilderdom.com/phd
# Highlight the general trend of a small drop between pre-program and first day (due to situational threat), the substantial increase during the program (probably inflated by post-group euphoria at program end), and the partial loss of gains during the follow-up period (typical of interventions). Of potential concern is that the 1st day to last day snapshot (typical of many evaluations) may over-estimate the amount of change during the program due to the initial depression and end-of-program euphoria. Thus, longitudinal research with a control group is preferable.

==See also==
* [[Advanced ANOVA/Data/LEQ]] - provides description of a downloadable LEQ database

==External links==
{{Hanging indent|1=Neill, J. T. (2008). ''Enhancing life effectiveness: The impacts of outdoor education programs''. [Doctoral dissertation, University of Western Sydney]. University of Western Sydney. https://researchers.westernsydney.edu.au/en/studentTheses/enhancing-life-effectiveness-the-impacts-of-outdoor-education-pro/}}
** Chapter 3 provides theoretical background to the LEQ
** Chapter 6 provides psychometric testing of the instrument
&lt;!--
* [http://wilderdom.com/leq Life Effectiveness Questionnaire] (Wilderdom) - Provides an overview and description of the instrumentation for potential users
--&gt;

[[Category:Life effectiveness]]</text>
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'''Life effectiveness''' is term which refers to the performance of generic personal and social life skills. Related terms include [[psychological resilience]] and [[practical intelligence]].

The '''Life Effectiveness Questionnaire''' provides a self-report instrument for assessing several purported life effectiveness skills, including:
# '''Time Management'''
# '''Social Competence'''
# '''Achievement Motivation'''
# '''Intellectual Flexibility'''
# '''Emotional Control'''
# '''Task Leadership'''
# '''Active Initiative'''
# '''Self Confidence'''

==Workshop activity==
# Present and discuss notion of life effectiveness - handout - for more info, see http://wilderdom.com/leq
## Related concepts: Practical intelligence, self-concept etc.
## Definitions of generic life effectiveness skills
## Describe the eight life effectiveness (LEQ-H) dimensions
# Present and discuss outdoor education results from http://wilderdom.com/phd
# Highlight the general trend of a small drop between pre-program and first day (due to situational threat), the substantial increase during the program (probably inflated by post-group euphoria at program end), and the partial loss of gains during the follow-up period (typical of interventions). Of potential concern is that the 1st day to last day snapshot (typical of many evaluations) may over-estimate the amount of change during the program due to the initial depression and end-of-program euphoria. Thus, longitudinal research with a control group is preferable.

==See also==
* [[Advanced ANOVA/Data/LEQ]] - provides description of a downloadable LEQ database

==External links==
{{Hanging indent|1=Neill, J. T. (2008). ''Enhancing life effectiveness: The impacts of outdoor education programs''. [Doctoral dissertation, University of Western Sydney]. University of Western Sydney. https://researchers.westernsydney.edu.au/en/studentTheses/enhancing-life-effectiveness-the-impacts-of-outdoor-education-pro/}}
* Chapter 3 provides theoretical background to the LEQ
* Chapter 6 provides psychometric testing of the instrument
&lt;!--
* [http://wilderdom.com/leq Life Effectiveness Questionnaire] (Wilderdom) - Provides an overview and description of the instrumentation for potential users
--&gt;

[[Category:Life effectiveness]]</text>
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'''Life effectiveness''' is term which refers to the performance of generic personal and social life skills. Related terms include [[psychological resilience]] and [[practical intelligence]].

The '''Life Effectiveness Questionnaire''' provides a self-report instrument for assessing several purported life effectiveness skills, including:
# '''Time Management'''
# '''Social Competence'''
# '''Achievement Motivation'''
# '''Intellectual Flexibility'''
# '''Emotional Control'''
# '''Task Leadership'''
# '''Active Initiative'''
# '''Self Confidence'''

==Workshop activity==
# Present and discuss notion of life effectiveness&lt;!-- - handout - for more info, see http://wilderdom.com/leq --&gt;
## Related concepts: Practical intelligence, self-concept etc.
## Definitions of generic life effectiveness skills
## Describe the eight life effectiveness (LEQ-H) dimensions
# Present and discuss outdoor education results which use the LEQ&lt;!-- from http://wilderdom.com/phd --&gt;
# Highlight the general trend of a small drop between pre-program and first day (due to situational threat), the substantial increase during the program (probably inflated by post-group euphoria at program end), and the partial loss of gains during the follow-up period (typical of interventions). Of potential concern is that the 1st day to last day snapshot (typical of many evaluations) may over-estimate the amount of change during the program due to the initial depression and end-of-program euphoria. Thus, longitudinal research with a control group is preferable.

==See also==
* [[Advanced ANOVA/Data/LEQ]] - provides description of a downloadable LEQ database

==External links==
{{Hanging indent|1=Neill, J. T. (2008). ''Enhancing life effectiveness: The impacts of outdoor education programs''. [Doctoral dissertation, University of Western Sydney]. University of Western Sydney. https://researchers.westernsydney.edu.au/en/studentTheses/enhancing-life-effectiveness-the-impacts-of-outdoor-education-pro/}}
* Chapter 3 provides theoretical background to the LEQ
* Chapter 6 provides psychometric testing of the instrument
&lt;!--
* [http://wilderdom.com/leq Life Effectiveness Questionnaire] (Wilderdom) - Provides an overview and description of the instrumentation for potential users
--&gt;

[[Category:Life effectiveness]]</text>
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  <page>
    <title>Social sustainability</title>
    <ns>0</ns>
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        <username>Atcovi</username>
        <id>276019</id>
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      <minor />
      <comment>cleanup</comment>
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{{Cleanup|need to organize page to better fit [[Wikiversity:Learning by doing]]}}
The concept of '''social sustainability''' should be thought of as a cultural and societal ideal which can be moved toward through design. It is best characterized by its ability to foster and sustain social interactions, as well as, create equitable opportunities for participation between the built environment and all of its users. By creating environments which do not disadvantage or disable its users; the environment becomes more socially sustainable. 

It is important to understand that the definition of what makes an environment socially sustainable can evolve as fast as the technology used in the processes of reaching for this form of sustainability. Designs and technology which may seem to be state of the art today could become outdated by advances in technology tomorrow. A good example of technology evolving behavior can be seen in the advances in communication technology. The internet is one prime example of a technological revolution which has completely changed the civilized world. We can expect unimaginable changes in technology to again reshape our future and the way we view the world. The principles set forth in this thesis should not be drastically affected by changes in technology. Creating principles which can transcend time and cannot be easily dated was a major goal of this undertaking. For this reason the guiding principles of this thesis will not be prescriptive designs but rather about the prescriptive methods for the creation of designs.

In order to create guiding principles for achieving socially sustainable college campuses, this thesis will explore what it takes to create environments which promote social interaction and equitable participation. Since social sustainability can be thought of as the outcome of other processes, it is important to explore the other processes which contribute to more socially sustainable environments. It is here where guiding principles for creating socially sustainable college campuses will be derived.&lt;ref&gt;Grimble, Michael. (June 2009). Social Sustainability and Collegiate Campuses. SUNY Buffalo, Buffalo, New York &lt;/ref&gt;

== References ==
{{Reflist}}</text>
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== ~2026-28792-52 ==

Please block [[Special:Contribs/~2026-28792-52]], vandalism. Appears to be same user as above. [[User:Tenshi Hinanawi|Tenshi Hinanawi]] ([[User talk:Tenshi Hinanawi|トーク]] • [[Special:Contributions/Tenshi Hinanawi|投稿記録]]) 17:38, 12 May 2026 (UTC)
: Blocked locally by Barras. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:46, 12 May 2026 (UTC)

== New User Exceeded New Page Limit  ==

Hello, my action of creating a Portal for Banjo learning (Portal:Banjo) was denied as I am a user whose had my account for a while but not written before today. My actions are constructive, but if you would rather me wait and let the system work as it intends to then that's okay too. 
--[[User:Kirby - Electrotechnics|Kirby - Electrotechnics]] ([[User talk:Kirby - Electrotechnics|discuss]] • [[Special:Contributions/Kirby - Electrotechnics|contribs]]) 01:13, 17 May 2026 (UTC)

:That is standard and the rate limit will fall off as you stay and edit; it has some pretty easy barriers to cross. I can create a blank [[Portal:Banjo]] if you want, but to be clear, portal pages are usually much broader topics that can help orient you to specific pages. Do you think there will be that many pages about the topic of banjo playing? If not, then you can just continue editing [[Banjo]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 03:36, 17 May 2026 (UTC)

== Disable [[Special:AbuseFilter/3]] ==

Please disable [[Special:AbuseFilter/3]] because it is redundant with [[m:Special:AbuseFilter/104|a global filter]] that disallows the same type of spam that filter 3 would have caught. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:06, 19 May 2026 (UTC)
: [[Special:AbuseFilter/history/3/diff/prev/454|Done]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:15, 19 May 2026 (UTC)

== [[One man's look at concept]] ==

Please semi-protect that page indefinitely due to vandalism from an unregistered vandal. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:52, 19 May 2026 (UTC)</text>
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== ~2026-28792-52 ==

Please block [[Special:Contribs/~2026-28792-52]], vandalism. Appears to be same user as above. [[User:Tenshi Hinanawi|Tenshi Hinanawi]] ([[User talk:Tenshi Hinanawi|トーク]] • [[Special:Contributions/Tenshi Hinanawi|投稿記録]]) 17:38, 12 May 2026 (UTC)
: Blocked locally by Barras. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:46, 12 May 2026 (UTC)

== New User Exceeded New Page Limit  ==

Hello, my action of creating a Portal for Banjo learning (Portal:Banjo) was denied as I am a user whose had my account for a while but not written before today. My actions are constructive, but if you would rather me wait and let the system work as it intends to then that's okay too. 
--[[User:Kirby - Electrotechnics|Kirby - Electrotechnics]] ([[User talk:Kirby - Electrotechnics|discuss]] • [[Special:Contributions/Kirby - Electrotechnics|contribs]]) 01:13, 17 May 2026 (UTC)

:That is standard and the rate limit will fall off as you stay and edit; it has some pretty easy barriers to cross. I can create a blank [[Portal:Banjo]] if you want, but to be clear, portal pages are usually much broader topics that can help orient you to specific pages. Do you think there will be that many pages about the topic of banjo playing? If not, then you can just continue editing [[Banjo]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 03:36, 17 May 2026 (UTC)

== Disable [[Special:AbuseFilter/3]] ==

Please disable [[Special:AbuseFilter/3]] because it is redundant with [[m:Special:AbuseFilter/104|a global filter]] that disallows the same type of spam that filter 3 would have caught. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:06, 19 May 2026 (UTC)
: [[Special:AbuseFilter/history/3/diff/prev/454|Done]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:15, 19 May 2026 (UTC)

== [[One man's look at concept]] ==

Please semi-protect that page indefinitely due to vandalism from an unregistered vandal. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:52, 19 May 2026 (UTC)

:I've protected the page, but I've decided to keep it to a year just because the vandalism seemed to have started in late 2025, and the longest protection imposed was just about 3 months. If vandalism persists after the protection, I'd be more than happy to protect it indefinitely. Thanks. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:24, 19 May 2026 (UTC)</text>
      <sha1>99hd8r9nzfyogt6gc935mgwdwor30tl</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Bots/Status/Archive</title>
    <ns>4</ns>
    <id>76212</id>
    <revision>
      <id>2810489</id>
      <parentid>2658287</parentid>
      <timestamp>2026-05-19T19:05:17Z</timestamp>
      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <comment>/* Leaderbot */ archive from [[Wikiversity:Bots/Status]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
      <origin>2810489</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="43782" sha1="2v0kr8u92jqfl256v4qbinzigi9uzl0" xml:space="preserve">Discussions are archived for review purposes. '''Please do not edit this page''', start a new discussion [[Wikiversity:Bots/Status|on the main page]] to discuss the topic further.

==[[User:Sebbot]]==
* '''Bot operator:''' [[User:Sebmol]]
* '''Aim of the bot:''' undo vandalism, rename categories
* '''Script used:''' [[w:WP:AWB]]
* '''Already used on, with bot status:''' [[w:de:Benutzer:Sebbot|German Wikipedia]]

{{done}} Bot was flagged:

 05:08, 3 October 2006 Cormaggio (Talk | contribs | block) granted bot status to User:Sebbot ‎(I trust you Seb :-))

==[[User:DraiconeBot]]==
* '''Bot operator:''' [[User:Draicone]]
* '''Aim of the bot:''' Simplify common tasks, fetch data, update counts, remove categories etc.
* '''Script used:''' Hmm... PHP? A mixture of [[w:WP:AWB]], pywikipedia, PHP with [http://sf.net/projects/snoopy snoopy], PHP using [http://en.wikiversity.org/w/query.php query.php], Manually operated using AJAXified [http://en.wikiversity.org/w/query.php query.php] etc.
* '''Already used on, with bot status:''' [[w:User:DraiconeBot|English Wikipedia]], the now defunct MWT wiki
{{Done}} [[User:sebmol|sebmol]] [[User talk:sebmol|&lt;sup&gt;?&lt;/sup&gt;]] 12:06, 31 October 2006 (UTC)

==[[User:Wherebot]]==
*'''nomination for bot status'''. This [[w:User:Wherebot|bot from Wikipedia]] watches for copyright violations. An example of how this could work at Wikiversity is at [[User:Wherebot]]. --[[User:JWSchmidt|JWSchmidt]] 16:39, 1 November 2006 (UTC)
:I'm willing to put the bot on as soon as it is approved. Here is the info (below): -- [[User:Where|Where]] 00:10, 31 October 2006 (UTC)
*'''Bot operator:''' [[w:User:Where]]
*'''Aim of the bot:''' detect copyright violations
*'''Script used:''' [[w:User:Wherebot]]
*'''Already approved on the English Wikipedia''
{{Done}}, assigning responsibility for the bot to [[User:JWSchmidt]]. [[User:sebmol|sebmol]] [[User talk:sebmol|&lt;sup&gt;?&lt;/sup&gt;]] 17:08, 1 November 2006 (UTC)

==[[User:HagermanBot]]==
* '''Bot Operator:''' [[User:Hagerman]]
* '''Aim of the bot:''' Automatically detects unsigned comments left on the talk namespaces as well as requested pages (indicated by placing the page in a special category). It places the &lt;nowiki&gt;{{unsigned}}&lt;/nowiki&gt; template after the comment. If the user leaves 2 unsigned comments within a 24 hour period, the bot leaves a message on the users talk page with information on how to properly sign a comment.
* '''Language used:''' Visual C# .NET
* '''Already approved on the English  Wikipedia''' See [[w:User:HagermanBot|HagermanBot]]

Please run the bot without flag for a couple of days so we can get a picture of how it works. [[User:sebmol|sebmol]] [[User talk:sebmol|&lt;sup&gt;?&lt;/sup&gt;]] 11:00, 15 December 2006 (UTC)
:Ok, the bot is running. Please contact me if you have any questions. Best, [[User:Hagerman|Hagerman]] 23:42, 15 December 2006 (UTC)

{{Done}} Granted bot status after test phase. [[User:sebmol|sebmol]] [[User talk:sebmol|&lt;sup&gt;?&lt;/sup&gt;]] 09:36, 2 January 2007 (UTC)

==[[User:MichaelBillingtonBot]]==
* '''Bot Operator:''' [[User:MichaelBillington]]
* '''Aim of the bot:''' Checks pages on the open proxy project for IPs which have been missed, then relists them on [[WV:OP]] for blocking. Also categorises subpages of the project and archives lists which contain only blocked IP addresses. (so it's really an all-round helper bot for the project)
* '''Language used:''' Visual Basic
* '''Already approved:''' English  Wikipedia under the same name (though that bot runs different software)
:I wrote this after I noticed I missed a large number of IP addresses on the first open proxy list (a thing best to avoid) [[User:MichaelBillington|Michael Billington]] ([[User talk:MichaelBillington|talk]] • [[Special:Contributions/MichaelBillington|contribs]]) 11:58, 6 January 2007 (UTC)

:'''Additional feature'''. I would like to have this bot also approved to post RSS feeds for Wikiversity blogs if a template is present. [[User:MichaelBillington|Michael Billington]] ([[User talk:MichaelBillington|talk]] • [[Special:Contributions/MichaelBillington|contribs]]) 07:24, 8 February 2007 (UTC)

Please run it without bot flag for about a week to test its abilities and check for bugs. [[User:sebmol|sebmol]] [[User talk:sebmol|&lt;sup&gt;?&lt;/sup&gt;]] 07:42, 8 February 2007 (UTC)

{{Done}} Granted bot status after test phase. [[User:sebmol|sebmol]] [[User talk:sebmol|&lt;sup&gt;?&lt;/sup&gt;]] 19:26, 25 March 2007 (UTC)

==[[User:ArchiveBot]]==
* '''Bot operator:''' [[User:Sebmol]]
* '''Aim of the bot:''' [[Template:auto archive|automatic archiving]]
* '''Script used:''' self-made
* '''Already used on, with bot status:''' [[w:de:Benutzer:ArchivBot|German Wikipedia]], [[:de:Benutzer:ArchivBot|German Wikiversity]]

{{done}} Bot was flagged:

 14:11, 25 March 2007 Sebmol (Talk | contribs | block) granted bot status to User:ArchiveBot ‎(for testing purposes)

==Commons Delinker==
[[User:CommonsDelinker]]

{{Done}} [[User:sebmol|sebmol]] [[User talk:sebmol|&lt;sup&gt;?&lt;/sup&gt;]] 20:07, 26 September 2007 (UTC)

==[[User:Crochet.david.bot|Crochet.david.bot]]==
* '''Bot operator:''' [[:fr:Utilisateur:Crochet.david]]
* '''Aim of the bot:''' interwiki
* '''Script used:''' interwiki.py of pywikipedia
* '''Already used on, with bot status:''' [[:fr:]] and [[:it:]]
--[[User:Crochet.david.bot|Crochet.david.bot]] 19:03, 10 August 2007 (UTC)

{{Done}} [[User:sebmol|sebmol]] [[User talk:sebmol|&lt;sup&gt;?&lt;/sup&gt;]] 20:07, 26 September 2007 (UTC)

==[[User:Mike's bot account|Mike's bot account]]==
*'''Bot operator:''' [[User:Mike.lifeguard|Mike.lifeguard]] (at en.wb: [[b:User:Mike.lifeguard]])
*'''Aim of the bot:''' So far, adding/removing categories/templates per request from [[User:SB_Johnny|SB Johnny]]. Any other tasks that can be done with [[w:WP:AWB|AWB]] can be requested. In the future, I can tag all untagged images and notify their uploaders.
*'''Script used:''' [[w:WP:AWB|AutoWikiBrowser]]
*'''Already used on:''' [[b:User:Mike's bot account|en.wb]], but without a bot flag. [[User:SB_Johnny|SB Johnny]] [http://en.wikibooks.org/w/index.php?title=User_talk:Mike.lifeguard&amp;diff=next&amp;oldid=984801 asked me] to request a bot flag for en.wv
'''&amp;ndash; [[User: Mike's bot account|MBA]]''' ( [[User Talk:Mike's bot account|talk]] | [[special:blockip/Mike's bot account|block]] ) 15:43, 26 September 2007 (UTC)

{{Done}} [[User:sebmol|sebmol]] [[User talk:sebmol|&lt;sup&gt;?&lt;/sup&gt;]] 20:07, 26 September 2007 (UTC)

== [[User:MichaelFreyTool]] ==
* '''Bot operator:''' [[User:MichaelFrey]]
* '''Aim of the bot:''' simple cleanup task, like resolve redirects
* '''Script used:''' [[w:WP:AWB]]
* '''Already used on, with bot status:''' [[b:de:User:MichaelFreyTool|de.wb]] and [[:de:User:MichaelFreyTool|de.wv]]
-- [[User:MichaelFrey|MichaelFrey]] 12:04, 27 January 2008 (UTC)
:This bot is running fine so far on de.WV (since it is just a piece of software and we know all Murphy's law: at least the edits I saw when asking the bot operator to do some changes). ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] [[Wikiversity:Chat|&lt;small&gt;Wikiversity:Chat&lt;/small&gt;]] 22:43, 15 March 2008 (UTC)
::{{Done}} ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] &lt;small&gt;uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])&lt;/small&gt; 20:58, 26 March 2008 (UTC)

==[[User:Computer]]==
* Bot operator: [[User:White Cat]] ([[:Commons:User:White Cat]]) - En-N, Tr-4, Ja-1
* List of botflags on other projects: Bot has a flag on wikimedia (meta,commons) wikipedia (ar, az, de, en, es, et, fr, is, ja, ku, nn, no, ru, sr, tr, uz, simple...) (See: [[m:User:White Cat#Bots]])
* Purpose: Interwiki linking, double redirect fixing, commons delinking (for cases where commonsdelinker fails)
--&lt;small&gt; [[User:White Cat|Cat]]&lt;/small&gt; &lt;sup&gt;[[User talk:White Cat|chi?]]&lt;/sup&gt; 17:42, 12 March 2008 (UTC)
:Bot uses pywikipedia and AWB framework. ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] [[Wikiversity:Chat|&lt;small&gt;Wikiversity:Chat&lt;/small&gt;]] 00:09, 16 March 2008 (UTC)

{{strike|So far missing basic criteria:
:*indicate on bot's user page:
::* which program / language is used (Pywikipedia, Ruby, javascript...)
::* who the owner is 
:*choose a name containing the word &quot;bot&quot; so that editors realize they are dealing with an automaton - see also [[Wikiversity_talk:Bots#Name_should_include_bot_.3F|Name should include bot ?]]
:Please also have a look at the other [[Wikiversity:Bots#Expectations|expectations]].}}
:&lt;s&gt;Anyway&lt;/s&gt;the projects where the bot is running seems quite impressive (though not each checked on the list from [[m:User:White Cat#Bots]], e.g. also de.WV is listed, but the [[:de:Wikiversity:Bots#Benutzer:Computer|status]] not given there yet). ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] [[Wikiversity:Chat|&lt;small&gt;Wikiversity:Chat&lt;/small&gt;]] 19:37, 15 March 2008 (UTC)
::Some feedback already at other Wikiversities about the request there:
::*[[:el:%CE%A8%CE%B7%CF%86%CE%BF%CF%86%CE%BF%CF%81%CE%AF%CE%B1#.CE.A3.CF.85.CE.B6.CE.AE.CF.84.CE.B7.CF.83.CE.B7|Greek Wikiversity]]
::*[[:fr:Wikiversité:Bot/Statut|French Wikiversity]]
::*[[:de:Wikiversity:Bots#Benutzer:Computer|German Wikiversity]]
::[[Special:Contributions/Computer|Contributions]] at this Wikiversity to see how the bot is operating, ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] [[Wikiversity:Chat|&lt;small&gt;Wikiversity:Chat&lt;/small&gt;]] 22:55, 15 March 2008 (UTC)
{{done}} --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | &lt;sup&gt;[[User_talk:SB_Johnny|{{font|color=green|talk}}]]&lt;/sup&gt; 10:53, 24 March 2008 (UTC)

== [[User:ArthurBot|ArthurBot]] ==
*[[Special:Contributions/ArthurBot|Contributions]]
*Owner: [[User:Mercy|Mercy]]
*Language: pywikipedia
*Functions: adding/correcting interwiki links
*Flags: cs

Hi, ArthurBot is a global bot with over 330,000 contribs. It's going to add interwiki links to newly created Czech pages and categories. Best regards, --[[User:Mercy|Mercy]] 10:14, 10 January 2009 (UTC)

{{done}} [http://en.wikiversity.org/w/index.php?title=Special%3ALog&amp;type=rights&amp;user=&amp;page=User%3AArthurBot&amp;year=&amp;month=-1]  --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 22:44, 16 January 2009 (UTC)

==[[User:It]] ([[tag]]bot)==
* '''Bot Operator:''' [&lt;kbd&gt;user=[[&amp;...|undefined]]&lt;/kbd&gt;]
* '''Aim of the bot:''' tracks and catalogs future resources logged through the use of the [[:Template:placeholder]] object, The '''&quot;[[&amp;...]]&quot; pseudo-namespace''' and the '''''[[Tag]]'''''® [[gaming engine]]
* '''Language used:''' [[Topic:XML|XML]]/[[UML]] with definitions and stubs for [[Topic:Perl]], [[Topic:Python]], [[Topic:PHP]], [[Topic:Smalltalk]], '''[[&amp;...]]''' 
* '''Under development:''' Proposed for use on any [[Topic:MediaWiki]]-driven site by the [[Wikia]][[Topic:Perl|Perl]] [http://perl.wikia.com group] (''[[Wikiversity:Help Wanted|Help Wanted]]'').

I'm not sure I quite understand the purpose of this bot or how it works. Can you post a simple demonstration that does not rely on Wikia? Feel free to use the bot account for that. [[User:sebmol|sebmol]] [[User talk:sebmol|&lt;sup&gt;?&lt;/sup&gt;]] 07:44, 8 February 2007 (UTC)

{{not done}}  There has been no response to the questions about the function of this bot.  --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 02:33, 26 January 2009 (UTC)

== [[User:Dinybot]]==

Hello, I would like to ask for the bot flag for the [[User:Dinybot|Dinybot]] robot. It is a classical global automatic interwiki robot which operates at all Wikiversities. Demonstrative editations are [[Special:Contributions/Dinybot|available]]. Also specification in English is [[User:Dinybot/Specification|available]]. Simple specifiation follows:

* '''Bot Operator''': [[User:Martin Kozák]]
* '''Aim of the Bot''': Periodical automated completing and keeping of the interwiki links of all projects.
* '''Language used''': [[w:Python|The Python]]
* '''Script used''': [[meta:Using_the_python_wikipediabot|Python Wikipedia Robot Framework]]
* '''Already used on''':
*# [[w:cs:Hlavní strana|Czech Wikipedia]] for automated typographic corrections, code cleanups and redirects replacing, &lt;small style=&quot;margin-left: 0.5em;&quot;&gt;(''Lightweight MediaWiki Robot Framework'', ''since 2006-05-11'')&lt;/small&gt;
*# [[q:cs:Wikicitáty:Hlavní strana|Czech Wikiquote]] for typographic correction, code cleanups and redirects replacing, &lt;small style=&quot;margin-left: 0.5em;&quot;&gt;(''Lightweight MediaWiki Robot Framework'', ''since November 2006'')&lt;/small&gt; 
*# [[q:User:Dinybot|Wikiquote]] projects for completing and keeping the interwiki links. &lt;small style=&quot;margin-left: 0.5em;&quot;&gt;(''Python Wikipedia Robot Framework'', ''since March 2007, in pending'')&lt;/small&gt; 

Thanks for Your complaisance.

--[[User:Martin Kozák|Martin Kozák]] 17:33, 27 March 2007 (UTC)

{{not done}}  This is a very old request, and it is not clear that the bot operator is still active.  Please resubmit the request if you are still interested in running the bot.  (and feel free to leave a note on the talk pages of the crats to insure a quicker response)  --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 02:33, 26 January 2009 (UTC)

=={{User|AdambroBot}}==

* '''Bot operator''': {{User|Adambro}}
* '''Automatic or manually assisted''': Semi-automatic
* '''Purpose of the bot''': Re-categorisation, other similar maintenance tasks as required
* '''Edit period(s)''': As required
* '''Programming language(s) (and API) used''': [[w:WP:AWB|AutoWikiBrowser]]
* '''Other projects that are already using this bot''': I have bot flagged accounts for similar maintenance tasks on the English Wikipedia, Commons, Meta. 
* '''Additional information''':  I'm an administrator on the English Wikipedia, the English Wikinews, and the Wikimedia Commons projects. I also operate a bot which uses the pywikipedia to make regular fully automated edits to the English and Finnish Wikinews projects which have amassed about 50,000 edits. See also [http://toolserver.org/~vvv/sulutil.php?user=AdambroBot AdambroBot on other projects].
{{done}} [[User:AdambroBot]] now has a [http://en.wikiversity.org/w/index.php?title=Special%3ALog&amp;type=rights&amp;user=&amp;page=User%3AAdambroBot&amp;year=&amp;month=-1 bot flag].  --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 13:32, 3 July 2009 (UTC)

=={{User|Mu301Bot}}==

* '''Bot operator''': {{User|Mu301}}
* '''Automatic or manually assisted''': manually assisted
* '''Purpose of the bot''': maintenance tasks
* '''Edit period(s)''': as needed
* '''Programming language(s) (and API) used''': [[meta:Using_the_python_wikipediabot|pywikipedia]]
* '''Other projects that are already using this bot''': none
* '''Additional information''': will also be used to develop a [[Pywikipediabot|learning project on using bots]]

:{{done}} After 6 weeks with no comments, I'm willing to flag for a trusted user. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]]  &lt;sup&gt;[[User_talk:SB_Johnny|{{font|color=green|talk}}]]&lt;/sup&gt; 00:23, 31 August 2009 (UTC)

=={{User|JackBot}}==

* '''Bot operator''': {{User|JackPotte}}
* '''Automatic or manually assisted''': Both.
* '''Purpose of the bot''': Cleaning the [[Special:DoubleRedirects|double redirections]] with the famous [[m:Pywikipediabot/redirect.py|redirect.py]].
* '''Edit period(s)''': Every day.
* '''Programming language(s) (and API) used''': AWB, Pywikipedia.
* '''Other projects that are already using this bot''': [http://toolserver.org/~vvv/sulutil.php?user=JackBot Several]
* '''Additional information''': If someone else absolutely wants to execute this script every day I'll stop this vote for my bot.

: Requested test run of bot [http://en.wikiversity.org/w/index.php?title=User_talk:JackPotte&amp;diff=598039&amp;oldid=505417 here]. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 01:35, 13 August 2010 (UTC)
::{{done}} [[User:JackPotte|JackPotte]] 04:59, 13 August 2010 (UTC)

::I've [http://en.wikiversity.org/w/index.php?title=Special%3ALog&amp;type=rights&amp;user=&amp;page=User%3AJackBot+&amp;year=&amp;month=-1 flagged JackBot]. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 05:21, 13 August 2010 (UTC)

=={{User|EdoBot}}==

* '''Bot operator''': {{User|EdoDodo}}
* '''Automatic or manually assisted''': Automatic
* '''Purpose of the bot''': Creating and maintaining [[:meta:Interlanguage links|interwiki links]].
* '''Edit period(s)''': Lots of runs.
* '''Programming language(s) (and API) used''': [[:meta:Pywikipediabot|Pywikipedia]] ([[:meta:Pywikipediabot/interwiki.py|interwiki.py]])
* '''Other projects that are already using this bot''': Approved on Simple English Wikipedia. Requested on French, Italian, and English Wikipedias.
* '''Additional information''': I realize that there's already an awful lot of interwiki bots out there, but [[wikt:redundancy|redundancy]] is never a bad idea, and one more can't hurt ;). I am an experienced bot operator, I am part of the [[:w:WP:BAG|Bot Approvals Group]] on enwp and operate three bots ([[:w:User:DodoBot|DodoBot]], [[:w:User:MessageDeliveryBot|MessageDeliveryBot]], [[:w:User:WelcomerBot|WelcomerBot]]) there, as well as [[commons:User:DodoBot|DodoBot]] on Commons. - [[w:User:EdoDodo|{{font|color=#21421E|face=&quot;Harrington&quot;|EdoDodo}}]] &lt;sup&gt;[[w:User talk:EdoDodo|&lt;span style=&quot;font-family:Times New Roman;color:#33dd44&quot;&gt;talk&lt;/span&gt;]]&lt;/sup&gt; 09:34, 6 September 2010 (UTC)

:Hello! Thank you for the offer to run a bot to help maintain wikiversity. As described in the [[Wikiversity:Bots]] page I would like to request that you run a limited test (just once and without the bot flag) to demonstrate the work that your bot will perform. Please note [[Wikiversity:Bots#Edit_rate_guidelines|the edit rate guidelines and the use of the MaxLag parameter]]. Let me know after you run the test.  --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 21:28, 8 September 2010 (UTC)
::[http://en.wikiversity.org/w/index.php?title=Wikiversity%3AWhat_is_Wikiversity%3F%2FEn&amp;action=historysubmit&amp;diff=609696&amp;oldid=606038 Sample edit] Thanks! - [[w:User:EdoDodo|{{font|color=#21421E|face=&quot;Harrington&quot;|EdoDodo}}]] &lt;sup&gt;[[w:User talk:EdoDodo|&lt;span style=&quot;font-family:Times New Roman;color:#33dd44&quot;&gt;talk&lt;/span&gt;]]&lt;/sup&gt; 18:36, 12 September 2010 (UTC)
:::{{done}} [http://en.wikiversity.org/w/index.php?title=Special%3ALog&amp;type=&amp;user=&amp;page=User%3AEdoBot&amp;year=&amp;month=-1&amp;hide_patrol_log=1] --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 01:08, 14 September 2010 (UTC)

=={{User|GedawyBot}}==

* '''Bot operator''': {{user|محمد الجداوي}}
* '''Automatic or manually assisted''': Automatic
* '''Purpose of the bot''': Interwiki
* '''Edit period(s)''': as needed
* '''Programming language(s) (and API) used''': Pywikibot
* '''Other projects that are already using this bot''': Global
* '''Additional information''':
--[[User:محمد الجداوي|محمد الجداوي]] 11:41, 20 October 2011 (UTC)
*{{support}} bot status - does a good job at Wikipedia. However, I would suggest that the operator creates an English language talk page here or on meta, because it can be difficult for users whose language has a Roman script to use the arabic Wikipedia talk page due to the right-to-left typing. --[[User:S Larctia|&lt;span style=&quot;font-family:Fraktur;font-size:100%;color:#003f3e;&quot;&gt;Simone&lt;/span&gt;]] 17:51, 20 October 2011 (UTC)
::Thanks. You can contact me on [[m:User talk:محمد الجداوي|my talk page on meta]].--[[User:محمد الجداوي|محمد الجداوي]] 02:31, 21 October 2011 (UTC)
*'''Question''' is this bot working on other WV projects currently? David.crochet's bot has been performing this task for several years now. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]]  &lt;sup&gt;[[User_talk:SB_Johnny|{{font|color=green|talk}}]]&lt;/sup&gt; 09:01, 1 November 2011 (UTC)
:Yes; My bot has bot flag on arabic WV (Although there is not much to do there; As it's a new site). I really want to help here.--[[User:محمد الجداوي|محمد الجداوي]] 11:59, 1 November 2011 (UTC)
*{{Comment}} - I'm noting that this bot was flagged on [http://meta.wikimedia.org/w/index.php?title=Steward_requests/Bot_status&amp;curid=31937&amp;diff=2810378&amp;oldid=2804094 21 other WMF wikis 18/11/2011]. The question remains though what does it do better or worse than David.crochet's bot? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:35, 6 November 2011 (UTC)
::I just want to run my interwiki bot here, I don't say i'm better or worth than anyone. You can test my edits.--[[User:محمد الجداوي|محمد الجداوي]] 13:41, 6 November 2011 (UTC)

SB_Johnny suggested I be contacted about this and I don't know why. What do people still want to know before granting the bot flag? What concerns are there to discuss? --[[User:Darklama|&lt;span style=&quot;background:DarkSlateBlue;color:white;padding:2px;&quot;&gt;&amp;nbsp;dark&lt;/span&gt;]][[User talk:Darklama|&lt;span style=&quot;background:darkslategray;color:white;padding:2px;&quot;&gt;lama&amp;nbsp;&lt;/span&gt;]] 12:58, 24 November 2011 (UTC)
:James and I don't know all that much about bots, so if you think the bot is fine, the bot is fine with me. Just looking for an expert nod. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]]  &lt;sup&gt;[[User_talk:SB_Johnny|{{font|color=green|talk}}]]&lt;/sup&gt; 18:17, 24 November 2011 (UTC)

{{not done}} This very old request is now archived. Please resubmit if you are interested in using a bot to contribe to Wikiversity.  --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 07:16, 9 November 2015 (UTC)

=={{User|Hazard-Bot}}==
* '''Bot operator''': {{User|Hazard-SJ}}
* '''Automatic or manually assisted''': Automatic
* '''Purpose of the bot''': Cleaning [[Wikiversity:Sandbox|the sandbox]]
* '''Edit period(s)''': I usually do this hourly, but as it has a much lower edit rate, I could change the frequency.
* '''Programming language(s) (and API) used''': Pywikipedia
* '''Other projects that are already using this bot''': [[w:|enwiki]], [[w:nl:|nlwiki]], [[mw:|mediawikiwiki]]
* '''Additional information''': As stated above, I believe hourly checks for this task might be irrelevant. I could do it, though, or I could do it at less frequent intervals (best for you, the community to decide, I think?). Also, it would be nice to have a page with the content it should be reset with (protection would be good to prevent vandalism), or at least a confirmation of the text it should be replaced with. &lt;b&gt;&lt;span style=&quot;border:2px solid;font-variant:small-caps&quot;&gt;[[User:Hazard-SJ|&lt;span style=&quot;background:#00008B;color:white&quot;&gt;&amp;nbsp;Hazard-SJ&amp;nbsp;&lt;/span&gt;]][[User talk:Hazard-SJ|&lt;span style=&quot;color:#00008B;background:red;&quot;&gt;&amp;nbsp;±&amp;nbsp;&lt;/span&gt;]]&lt;/span&gt;&lt;/b&gt; 23:13, 21 June 2012 (UTC)
::Hello, and thank you for your offer to help maintain Wikiversity! I apologize for noticing this earlier. In the past I have run a bot to &quot;rake&quot; the sandbox once per week, but my bot is not currently editing this page. We do have a template at {{tl|Please leave this line alone (sandbox heading)}} and you can see how it is used by viewing the wiki source at this revision: [http://en.wikiversity.org/w/index.php?title=Wikiversity:Sandbox&amp;oldid=951484]. A few questions:
::*Do you plan to use the bot for other tasks?
::*Are you interested in becoming active at any of the Wikiversity learning projects?
:: --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:43, 8 October 2012 (UTC)

:::As for other tasks, not immediately, but I'd request here if I find something else that I could do with the bot. As for becoming active on Wikiversity, hmm ... not specifically. I edit on other wikis fairly actively, though. I'm a member of the [[m:SWMT|SWMT]] (and am a global rollbacker, by extension), merely suggesting that I've had many crosswiki experiences. I run bots on other projects, including sandbots on three other wikis, so that should be okay. &lt;b&gt;&lt;span style=&quot;border:2px solid;font-variant:small-caps&quot;&gt;[[User:Hazard-SJ|&lt;span style=&quot;background:#00008B;color:white&quot;&gt;&amp;nbsp;Hazard-SJ&amp;nbsp;&lt;/span&gt;]][[User talk:Hazard-SJ|&lt;span style=&quot;color:#00008B;background:red;&quot;&gt;&amp;nbsp;±&amp;nbsp;&lt;/span&gt;]]&lt;/span&gt;&lt;/b&gt; 01:52, 23 January 2013 (UTC)

{{not done}} This very old request is now archived. Please resubmit if you are interested in using a bot to contribe to Wikiversity.  --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 07:17, 9 November 2015 (UTC)

=={{User|MaintenanceBot}}==
* '''Bot operator''': {{User|Dave Braunschweig}}
* '''Automatic or manually assisted''': Manually assisted for now.
* '''Purpose of the bot''': A variety of maintenance tasks, starting with identifying unlicensed files and adding notifications to the file and the user.
* '''Edit period(s)''': As needed.
* '''Programming language(s) (and API) used''': Windows PowerShell
* '''Other projects that are already using this bot''': None
* '''Additional information''': Licensing information for contributed files is currently unmanaged.  The scope of the problem is beyond the time and abilities of human efforts alone.  A bot is necessary to address the problem.  -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 11:42, 30 October 2013 (UTC)
*: Hi Dave, Thankyou for your diligent attention to unlicensed image uploads etc. I've enabled this bot. Would you mind listing the specific actions it should perform on the bot's user page? Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:07, 10 November 2013 (UTC)
*::{{Done}} I'll also post an example on the [[User_talk:MaintenanceBot | MaintenanceBot talk]] page when it's ready.  Thanks!  -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:28, 10 November 2013 (UTC)

{{done}} This request was [https://en.wikiversity.org/w/index.php?title=Special%3ALog&amp;type=rights&amp;user=&amp;page=MaintenanceBot&amp;year=&amp;month=-1&amp;tagfilter= granted] by [[User:Jtneill]] on 10 November 2013. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 07:20, 9 November 2015 (UTC)

== RileyBot ==
* '''Bot name''': {{User|RileyBot}}
* '''Bot operator''': {{User|Riley Huntley}}
* '''Automatic or manually assisted''': Automatic
* '''Purpose of the bot''':  Clean [[Wikiversity:Sandbox]] as the bot tasked to due so hasn't done it in nearly a month
* '''Edit period(s)''': Every six hours, or as otherwise requested
* '''Programming language(s) (and API) used''': Python, pywikipedia
* '''Other projects that are already using this bot''': amgwiki, commonswiki, enwiki, enwikiquote, enwikivoyage, eswikivoyage, frwiivoyage, hewikisource, sawiki, simplewiki, sourceswiki, thwiki, ttwiktionary
* '''Additional information''': Bot would delay if the page has been edited within an hour. -[[User:Riley Huntley|Riley Huntley]] [[Meta:SWMT|(SWMT)]] 22:52, 4 February 2016 (UTC)


:{{ping|Riley Huntley}} Hello, and thank you for offering to contribute to Wikiversity! Usually we leave these requests open for about a week in case the community has any questions. I don't expect any for a routine request like this. I was running the bot that watched the sandbox but I set it at a very slow clean of once per week. The more frequent clean with the delay you suggest sounds like a better method to prevent interruption of testing. I suspect most editors now test in the userspace sandbox in any case. I'll watch for comments here and check back in a little while. For now, go ahead and start the bot (w/o the flag) to make some periodic edits so I can see the revision history. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 03:00, 5 February 2016 (UTC)

:Please also create [[User:RileyBot]] with info about the programming language and linking to the operator page. Include either {{tl2|User bot}} or add the bot userpage to [[:Category:Wikiversity bots]]. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 03:06, 5 February 2016 (UTC)
::{{not done}} - Very stale request. Requests made by mikeu were not act up (creating info), bot didn't run, and the owner of the request hasn't even commented on his/her own request. Please make a new request if you are still interested in your bot becoming an official, flagged bot here at Wikiversity... other than that, this request won't be accepted. ---[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:28, 20 November 2016 (UTC)

== Texvc2LaTeXBot ==
* '''Bot name''': {{User|Texvc2LaTeXBot}}
* '''Bot operator''': {{User|Salix alba}}
* '''Automatic or manually assisted''': automatic but closely monitored 
* '''Purpose of the bot''': to replace deprecated syntax in the LaTex mathematical syntax
* '''Edit period(s)''': A one off run affecting 2233 pages.
* '''Programming language(s) (and API) used''': python/piwikibot 
* '''Other projects that are already using this bot''': enwiki, dewiki many other wiki across the whole of WikiMedia sites.
* '''Additional information''': All ready running on the English wikipedia at [[en:User:Texvc2LaTeXBot]]. Has sucessfully run on many other wikis, the update plan is described in [[mw:Extension:Math/Roadmap]]. The bot will apply 10 regexps replacing deprecated syntax inside {{tag|math}} tags. The code has been tested to eliminate false positives. The bot is operated by Salix alba on English language wikis and [[meta:User:Debenben]] on German and other language wikis.

:{{done}} {{ping|Salix alba}} I set it for a one year expiry only due to the limited scope of the request but would be happy to extend it if you have further tasks in the future. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 23:26, 11 May 2019 (UTC)

== DannyS712 bot ==
* '''Bot name''': {{User|DannyS712 bot}}
* '''Bot operator''': {{User|DannyS712}}
* '''Automatic or manually assisted''': automatic (manually triggered)
* '''Purpose of the bot''': Tag pages that are uncategorized with {{tl|Uncategorized}}
* '''Edit period(s)''': One time run for 4000+ pages, then if needed will run on occasion (but rarely)
* '''Programming language(s) (and API) used''': AWB
* '''Other projects that are already using this bot''': No other wikis for this task, but lots of tasks on enwiki (and one on enwikibooks)
* '''Additional information''': Please ping when replying.

Thanks, --[[User:DannyS712|DannyS712]] ([[User talk:DannyS712|discuss]] • [[Special:Contributions/DannyS712|contribs]]) 04:52, 12 September 2019 (UTC)

:{{At|DannyS712}} Please start a discussion in the [[Wikiversity:Colloquium]] to confirm which pages should be tagged as {{tl|Uncategorized}}. Looking at [[Special:UncategorizedPages]], I don't see any problems tagging main space landing pages, but tagging subpages doesn't appear to add value. Perhaps they could be updated with {{tlx|CourseCat}}, but the community should have an opportunity to discuss this first. Thanks! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 15:01, 12 September 2019 (UTC)
::I agree with Dave about the CourseCat for subpages and Uncategorized for top level pages. I'd like to hear what the community thinks is best. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 15:57, 24 September 2020 (UTC)

{{not done}} due to lack of reply. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 22:24, 14 January 2021 (UTC)

== &lt;s&gt;Listeriabot&lt;/s&gt; WikiJournalBot ==
===Listeriabot===
* '''Bot name''': {{User|ListeriaBot}}
* '''Bot operator''': {{User|Magnus Manske}} (Though {{User|Evolution and evolvability}} will supervise on this wiki)
* '''Automatic or manually assisted''': Automatic 
* '''Purpose of the bot''': Formatting up wikitables based on sparql queries
* '''Edit period(s)''': 1 week test, (if no problems, indefinitely)
* '''Programming language(s) (and API) used''':
**Languages = Rust
**API = https://www.wikidata.org/w/api.php
**Repository = [https://github.com/magnusmanske/listeria_rs magnusmanske/listeria_rs])
* '''Other projects that are already using this bot''': commons, meta, wikidata, wikispecies, ,43 languages of wikipedia (incliuding english) - [https://listeria.toolforge.org/botstatus.php full list]
* '''Additional information''': The bot creator ({{U|Magnus Manske}}) isn't good at replying to talkpage comments or emails, but does seem to fix raised issues reasonably rapidly. My main discomfort is that I don't speak Rust, so can't directly troubleshoot any issues. I've tested the bot quite extensively on other wikis and have found it to be very reliable (at the very least, it has only ever edited pages with the {{tlx|wikidata list}} template, so there is minimal risk of disruption). My very minimal test here was to run [https://listeria.toolforge.org/index.php?action=update&amp;lang=en&amp;project=wikiversity&amp;page=User:Evolution_and_evolvability/sandbox this command] on [[User:Evolution_and_evolvability/sandbox|my sandbox]] which caused {{U|ListeriaBot}} to apply for an account, but it has not attempted to make an edit. [[User:Evolution and evolvability|T.Shafee(Evo&amp;#65120;Evo)]]&lt;sup&gt;[[User talk:Evolution and evolvability|talk]]&lt;/sup&gt; 11:13, 24 February 2021 (UTC)

{{At|Evolution and evolvability}} Can we get a test of the bot here that we can see, or do we need to enable the bot flag first? -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:15, 24 February 2021 (UTC)
:{{re|Dave Braunschweig}} I think the bot flag needs to be applied for it to run a test. You can see a test case below:
:*[[User:Evolution and evolvability/sandbox|Testcase on Wikiversity]] (click &quot;Update now&quot; to test)
:*[https://meta.wikimedia.org/w/index.php?title=User:Evolution_and_evolvability/sandbox&amp;oldid=21145459 Testcase on Meta] (showing what the result should be)
:[[User:Evolution and evolvability|T.Shafee(Evo&amp;#65120;Evo)]]&lt;sup&gt;[[User talk:Evolution and evolvability|talk]]&lt;/sup&gt; 22:31, 24 February 2021 (UTC)

{{At|Evolution and evolvability}} [[User:ListeriaBot]] is now tagged as a bot with a one-week expiration. Please update here if the test is successful and you want to go ahead with indefinite bot status. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:48, 24 February 2021 (UTC)

{{not done}} due to no activity in over one year. This is done without prejudice and we welcome a new application to suggest improvements to Wikiversity. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 22:19, 1 June 2022 (UTC)

== Arbota ==
* '''Bot name''': {{User|Arbota}}
* '''Bot operator''': {{User|Bocardodarapti}}
* '''Automatic or manually assisted''': halfautomatic (manually triggered)
* '''Purpose of the bot''': at the moment: helping me with generating pages of translated exercises from German wikiversity. (The Wikidatalinks I do via quickstatements on toolforge).
* '''Edit period(s)''': once in a while
* '''Programming language(s) (and API) used''':Pywikibot running on jupiter. 
* '''Other projects that are already using this bot''': German wikiversity. I started the bot some weeks ago. 
* '''Additional information''': I only use it for help within 'my pages' and always control the result.  

[[User:Bocardodarapti|Bocardodarapti]] ([[User talk:Bocardodarapti|discuss]] • [[Special:Contributions/Bocardodarapti|contribs]]) 06:39, 28 September 2022 (UTC)

:{{not done}} due to no activity in over one year. This is done without prejudice and we welcome a new application to suggest improvements to Wikiversity. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:00, 1 October 2024 (UTC)

===WikiJournalBot===
* '''Bot name''': {{User|WikiJournalBot}}
* '''Bot operator''': {{User|Evolution and evolvability}} &amp; {{User|Octfx}}
* '''Automatic or manually assisted''': Automatic 
* '''Purpose of the bot''': Formatting up lists of wikijournal articles based on sparql queries
* '''Edit period(s)''': 2 week test, (if no problems, indefinitely)
* '''Programming language(s) (and API) used''':
**Languages = PHP
**API = https://www.wikidata.org/w/api.php
**Repository = [https://github.com/octfx/WikiJournalBot octfx/WikiJournalBot]
* '''Other projects that are already using this bot''': none
* '''Additional information''': Having failed to get in contact with {{U|Magnus Manske}} to update listeriabot, I asked {{User|Octfx}}to put together a stripped-down version that is specialised for the wikijournal tasks we need. In particular, automating the lists:
** Currently: [[WikiJournal_of_Medicine]], [[WikiJournal_of_Science]], [[WikiJournal_of_Humanities]]
** Eventually: [[WikiJournal_User_Group/Potential_upcoming_articles]]
** Eventually: [[WikiJournal_User_Group/Editors]]
As with listeriabot, it only edits pages between the templates {{tlx|WikiJournalBotList}} and {{tlx|ListEnd}}, see sandbox tests [[User:Octfx/Volume_4_Issue_1]]. [[User:Evolution and evolvability|T.Shafee(Evo&amp;#65120;Evo)]]&lt;sup&gt;[[User talk:Evolution and evolvability|talk]]&lt;/sup&gt; 23:10, 16 March 2022 (UTC)

:{{At|Evolution and evolvability|Octfx}} Can someone create the WikiJournalBot user account and perform an edit under that account? So far, it doesn't appear to exist. Thanks! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 13:07, 17 March 2022 (UTC)
::@[[User:Dave Braunschweig|Dave Braunschweig]] Ah yes, rather easier to apply permissions to an existing account. Done. [[User:Evolution and evolvability|T.Shafee(Evo&amp;#65120;Evo)]]&lt;sup&gt;[[User talk:Evolution and evolvability|talk]]&lt;/sup&gt; 22:31, 17 March 2022 (UTC)
:::{{At|Evolution and evolvability|Octfx}} I followed the previous approach, bot for a week, presuming that was the intent. Let us know how testing goes. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 22:41, 17 March 2022 (UTC)
::::{{At|Dave Braunschweig|Evolution and evolvability}} I've created an OAuth consumer at meta and ran the bot through toolforge. Successful edits were made to the [[Wikiversity:Sandbox|Sandbox]] and two of my user pages [[User:Octfx/sandbox2|here]] and [[User:Octfx/Volume_4_Issue_1|here]]. --[[User:Octfx|Octfx]] ([[User talk:Octfx|discuss]] • [[Special:Contributions/Octfx|contribs]]) 09:51, 18 March 2022 (UTC)
:{{At|Evolution and evolvability|Octfx}} I'm fine with running tests as you have recently been doing and I highly encourage you to continue. Please note: &quot;When testing, bot operators must delay 60 seconds between edits.&quot; (see: [[WV:BOT]]) If you need assistance with this Dave or I can provide help with how to 'throttle' the edit rate to comply with this guideline. This makes it easier for custodians and 'crats to review the edits as they occur. Thank you so much for your contributions to improving Wikiversity, it is greatly appreciated. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 22:40, 1 June 2022 (UTC)
::Forgot to ping {{at|Dave Braunschweig}} for input. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 22:47, 1 June 2022 (UTC)
:: {{at|Mu301}} no problem, I’ll set the bot back to one edit per minute. The current delay was set while the bot flag was temporarily active, I forgot to change it back! —-[[User:Octfx|Octfx]] ([[User talk:Octfx|discuss]] • [[Special:Contributions/Octfx|contribs]]) 06:49, 2 June 2022 (UTC)
:::{{done}} - Great, we very much appreciate your contributions to Wikiversity. Thank you for improving our site and let us know if you need additional tools to improve the quality of our website. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 07:56, 2 June 2022 (UTC)

I don't know the inner workings of this bot, but it often shuffles the order of publications and made them out of publishing sequence[https://en.wikiversity.org/w/index.php?title=WikiJournal_of_Medicine/Volume_7_Issue_1&amp;diff=2417507&amp;oldid=2417404] [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 16:52, 23 August 2022 (UTC)

:@[[User:OhanaUnited|OhanaUnited]] You need to notify the bot owner or leave a message the bot's talk page. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 20:55, 23 August 2022 (UTC)

== WorkmarketBot ==
* '''Bot name''': {{User|WorkmarketBot}}
* '''Bot operator''': {{User|Evolution and evolvability}} &amp; {{User|Ecsussman}}
* '''Automatic or manually assisted''': Automatic
* '''Purpose of the bot''': Syncronising a list of tasks between [[WikiJournal_User_Group/Technical_editors/tasks]] and the external software [https://www.workmarket.com/ Workmarket]. Essentially performing two types of edit:
** Adding new rows to the table on that page (if item was added in Workmarket)
** Adding an &quot;Assignment ID&quot; to a column (if item added on-wiki and imported to workmarket)
* '''Edit period(s)''': indefinitely
* '''Programming language(s) (and API) used''':
** APIs = https://en.wikiversity.org/w/api.php and https://employer-api.workmarket.com
** language and repo = Java
* '''Other projects that are already using this bot''': none
* '''Additional information''': We've done a few test edits to check its functionality on a [[WikiJournal User Group/Technical editors/tasks/sandbox|sandbox]]. It would only be editing a single wiki page, so its activities are also very contained.
Note for transparency: I have also assigned it IP block exempt status during these tests since it was flagged by [[m:User_talk:Tks4Fish|Tks4Fish]]. I think this is reasonable, but please let me know if that's against proccedure to IP block exempt a both that I myself am applying permission for. [[User:Evolution and evolvability|T.Shafee(Evo&amp;#65120;Evo)]]&lt;sup&gt;[[User talk:Evolution and evolvability|talk]]&lt;/sup&gt; 11:48, 9 June 2023 (UTC)

:{{done}} - Bot permission added. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 21:51, 26 June 2023 (UTC)

== Leaderbot ==
* '''Bot name''': {{User|Leaderbot}}
* '''Bot operator''': {{User|Leaderboard}}
* '''Automatic or manually assisted''': Automatic
* '''Purpose of the bot''': [[meta:Global_reminder_bot]]
* '''Edit period(s)''': Daily (see below though)
* '''Programming language(s) (and API) used''': Python
* '''Other projects that are already using this bot''': Wikifunctions and some projects that do not require approval for bots that do not require a bot flag. See [[meta:Global reminder bot/global]] for the full list.
* '''Additional information''': I don't expect this to be used all that much, but the bots page requires approval regardless. This will ''not'' edit in a way requiring a bot flag. [[User:Leaderboard|Leaderboard]] ([[User talk:Leaderboard|discuss]] • [[Special:Contributions/Leaderboard|contribs]]) 18:55, 22 August 2024 (UTC)

:: {{ping|Leaderboard}} Thanks for the ping on my talk page. [https://en.wikiversity.org/wiki/Special:Log?type=rights&amp;user=&amp;page=Leaderbot Leaderbot user rights activated]. Could you perhaps update here: [[Wikiversity:Bots#Currently flagged bots]]? Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:20, 19 September 2024 (UTC)
:::@[[User:Jtneill|Jtneill]] {{done}} [[User:Leaderboard|Leaderboard]] ([[User talk:Leaderboard|discuss]] • [[Special:Contributions/Leaderboard|contribs]]) 04:55, 19 September 2024 (UTC)</text>
      <sha1>2v0kr8u92jqfl256v4qbinzigi9uzl0</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Request custodian action/Header</title>
    <ns>4</ns>
    <id>83382</id>
    <revision>
      <id>2810499</id>
      <parentid>2809902</parentid>
      <timestamp>2026-05-19T21:59:08Z</timestamp>
      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <comment>Dark mode support.</comment>
      <origin>2810499</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4620" sha1="gj4oqaugufhgimsxkgqwwqkus0b2dwp" xml:space="preserve">&lt;includeonly&gt;__NEWSECTIONLINK__{{Nobots}}&lt;/includeonly&gt;{{RoundBoxTop|theme=14}}
{{Portal-head2|2=Welcome}}
&lt;hr&gt;
{| class=&quot;wikitable&quot; style = &quot;float:right; margin-left: 1em&quot;
! Custodian requests !! Entries
|-
| [https://en.wikiversity.org/w/index.php?title=Wikiversity:Request_custodian_action/Header&amp;action=purge Purge cache]
|-
| {{ #ifexpr: {{PAGESINCAT:Wikiversity protected edit requests}} &gt; 0 | style=&quot;background: var(--background-color-destructive-subtle, #ff9090); color: inherit;&quot; |}} | [[:Category:Wikiversity protected edit requests|Edit protected page]] || {{ #ifexpr: {{PAGESINCAT:Wikiversity protected edit requests}} &gt; 0 | style=&quot;background: var(--background-color-destructive-subtle, #ff9090); color: inherit;&quot; |}} | {{PAGESINCAT:Wikiversity protected edit requests}}
|-
| {{ #ifexpr: {{PAGESINCAT:Candidates for speedy deletion}} &gt; 0 | style=&quot;background: var(--background-color-destructive-subtle, #ff9090); color: inherit;&quot; |}} | [[:Category:Candidates for speedy deletion|Speedy deletion]] || {{ #ifexpr: {{PAGESINCAT:Candidates for speedy deletion}} &gt; 0 | style=&quot;background: var(--background-color-destructive-subtle, #ff9090); color: inherit;&quot; |}} | {{PAGESINCAT:Candidates for speedy deletion}}
|-
| {{ #ifexpr: {{PAGESINCAT:Pending deletions}} &gt; 0 | style=&quot;background: var(--background-color-destructive-subtle, #ff9090); color: inherit;&quot; |}} | [[:Category:Pending deletions|Pending deletions]] || {{ #ifexpr: {{PAGESINCAT:Pending deletions}} &gt; 0 | style=&quot;background: var(--background-color-destructive-subtle, #ff9090); color: inherit;&quot; |}} | {{PAGESINCAT:Pending deletions}}
|-
| {{ #ifexpr: {{PAGESINCAT:Requests for Deletion}} &gt; 0 | style=&quot;background: var(--background-color-destructive-subtle, #ff9090); color: inherit;&quot; |}} | [[:Category:Requests for Deletion|Requests for Deletion]] || {{ #ifexpr: {{PAGESINCAT: Requests for Deletion}} &gt; 0 | style=&quot;background: var(--background-color-destructive-subtle, #ff9090); color: inherit;&quot; |}} | {{PAGESINCAT: Requests for Deletion}}
|-
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  <page>
    <title>Algorithms and Data Structures</title>
    <ns>0</ns>
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      <minor />
      <comment>/* See also */ format</comment>
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      <text bytes="2290" sha1="lcf5cf7firops9yvvx5ypb1wuiwjcic" xml:space="preserve">Algorithm -- briefly, is a step-by-step instruction that as a whole complete a process. Usually this process solves problems
such as sorting , finding data, and many other problems. It can be thought of as a recipe, with each ingredient contributing 
to the whole food, thus solving your problem of hunger :)
__NOTOC__
==Examples of Algorithms==

Linear searching algorithm : A linear searching has O(n) time complexity. Lets step through a process of creating an algorithm for
this problem.

'''Steps:'''
#We need a way to check through every element in the data. This can be done through a loop. A loop (ex, for loop , while loop) enables the programmer to run a block of code multiple times
#Each iteration we need to check if the value that we are searching for matches. If so then return that index.
#handle error, like what happens if the value we are searching for is not there ?

Since we have the step-by-step process written. We are able to effectively write an algorithm that follows the process above.

This function is written in C++.

 int searchLinearly( int * Array, int MaxSize, int searchValue)
 {
    //Step # 1 - loop through every element if needed
    for(int i = 0; i &lt; MaxSize; i++)
    {
        if( Array[i] == searchValue ) // Step # 2 -- search value
            return i;
    } 
       
    return -1; //Step # 3 -- handle error if value not found
 } 


So as you can see there is a step-by-step process in creating an algorithm. The above code could be written in template version but I chose not to for simplicity sake. 

==See also==

* [[Algorithms|Topic:Algorithms]] 
{{wikibooks|Algorithms}} 

== Learning readings==
===Wikipedia===
* [[w:list of algorithms|List of algorithms]] 
* [[w:Data structure|Data structure]]
* [[w:Data type|Data type]]
* [[w:List of data structures|List of data structures]]
* [[w:Category:Algorithms and data structures|Category:Algorithms and data structures]]
* [[w:Algorithms + Data Structures = Programs|Algorithms + Data Structures = Programs]] (book)
* [[w:List of terms relating to algorithms and data structures|List of terms relating to algorithms and data structures]]
* [[w:Algorithm|Algorithm]]
* [[w:Search data structure|Search data structure]]

== Data Structure Article ==
 Coming soon.

[[Category:Algorithms]]</text>
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  <page>
    <title>Mauchly's sphericity test</title>
    <ns>0</ns>
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      <id>2810420</id>
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      <text bytes="785" sha1="mqimaps2kir627mc0g01aelpoqokp5u" xml:space="preserve">{{statistics}}
'''Mauchly's sphericity test''' is used to test the assumption of '''sphericity''' in within-subject [[ANOVA]]s. This assumption is that: '''the difference scores between each within-subject variable have similar variances'''.

If Mauchly's sphericity test is significant (''p'' &lt; .05), then sphericity should not be assumed and adjustments to the degrees of freedom should be made using an adjustment such as:
# Greenhouse-Geisser
# Huyn-Feldt
# Lower-bound

Greenhouse-Geisser is commonly used and recommended.

==See also==
* [[w:Mauchly's sphericity test|Mauchly's sphericity test]] (Wikipedia)

==External links==
* [http://homepages.gold.ac.uk/aphome/spheric.html An introduction to sphericity]
* [http://www.microbiologybytes.com/maths/spss4.html ANOVA with SPSS]</text>
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  <page>
    <title>Qualitative research methods</title>
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      <text bytes="1570" sha1="gtuxtfgmyv37qas2sxvqy29r5vl559n" xml:space="preserve">{{psychology}}
Qualitative methods in sociological research refer to distinctive types of research activities: participant observation, intensive interviewing, and focus groups. The three qualitative designs differ but also share many similar characteristics that distinguish them from quantitative methods of research.  Qualitative researchers begin their research with an exploratory research question (for many times there isn't sufficient data to formulate a structured and specific goal).  And once started they make sure to pay attention to the social context in which social phenomena occur, human subjectivity and  how they themselves can influence any situation.

== Participant observation ==
 
'''Participant observation''' is a method for gathering data that involves developing a relationship with people while they go about their daily, normal activities.  It is a means for seeing the social world as the research subjects see it, in its totality, and for understanding subjects' interpretations of that world (Wolcott, 1995:66).

For more information, see [[w:Participant observation|Participant observation]] (Wikipedia)

== Intensive Interviewing ==

is a method in which the researcher seeks in-dept information from their interviewee's feelings, experiences, and perceptions.

== Focus Groups ==

is a method in where the researcher seeks to encourage discussion among participants about a certain topic of interest.

== References ==

Wolcott, Harry F. 1995. ''The Art of Fieldwork''. Walnut Creek, CA: AltaMira Press.

[[Category:Research methods]]</text>
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  <page>
    <title>Intrusion detection</title>
    <ns>0</ns>
    <id>96421</id>
    <revision>
      <id>2810415</id>
      <parentid>575067</parentid>
      <timestamp>2026-05-19T12:57:55Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>cleanup</comment>
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      <text bytes="2882" sha1="swslvx75maiepevhhuybmk16ldocydd" xml:space="preserve">{{uncategorized}}
{{cleanup|seems to be an undeveloped draft}}
'''Abstract'''

With the rapid development of Internet, it is an important task to ensure that college students accessing the Internet in a healthy way. This paper discusses the monitoring of user behavior by means of SNORT software in order to establish a campus network security monitoring system. 

Key words: Campus Network; SNORT; security; 

'''I.  INTRODUCTION''' 
Developed in 1969, ARPANet, the first form of internet, was initially used for military purposes. 
In 1993, it started being used for other non-military applications and business and hence  entered a stage of rapid development. In 1994, India introduced the Internet; in only a few years, the application of internet technology and services has grown tremendously, providing more and more new services.  
  
At the same time, the Internet has also brought forth a multitude of media and information to the public, both beneficial and harmful. Users can view illegal sites of violence, pornographic material, anti-government, and other undesirable content, creating detrimental effects to the user’s mental and physical health. Thus, it is imperative to study, develop, and employ measures which create a real-time security monitoring system capable of controlling and protecting the information available in order to ensure the healthy growth of college students and also the smooth and safe operation of the school network.   

This paper investigates the capabilities of SNORT to analyze those specific sensitive vocabularies of web contents and to monitor in real-time any unusual behaviors so as to establish a security system for campus network. 

'''II. THE TECHNOLOGY OF SNORT''' 

Snort is a lightweight network intrusion detection software, which is based on the network packet sniffer and logging tools in the lib pcap. 

Snort is scalable and portable. Developed in C, this open source software is free to use by any organization or individual.   

Snort is a software based on feature detection by which suspicious data packets are 
inspected and analyzed according to pre-defined rules; Once a rule is triggered, it generates an alert message.   Snort can generate reporting messages in real-time, each of which can be directed to user defined destination; it also delivers WinPopup messages to Windows client programs by using SAMBA protocols.   

Snort is comprised mainly of the following four components: 
#1. Data collection module that collects status data and feeds the data to the detection module. 
2. Detection module that detects and analyzes any intrusion behaviors and sends real-time alert messages. 
3. Knowledge database that provides necessary supporting data information.
4. Control module that responds automatically or manually to alert messages.

What are the ways to use the same for the Campus Network System?</text>
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  <page>
    <title>Auditory Cortical Evolution</title>
    <ns>0</ns>
    <id>97885</id>
    <revision>
      <id>2810413</id>
      <parentid>1864616</parentid>
      <timestamp>2026-05-19T12:56:52Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>PROD</comment>
      <origin>2810413</origin>
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      <text bytes="1141" sha1="bsche9a1mtfvyjczxr489lbkur3pfwl" xml:space="preserve">{{Prod|Undeveloped project since 2010}}
Humans have developed an advanced method of communication, speech.  This project aims to understand the basic anatomical, physiological, and molecular changes that have occurred in the auditory system across a range of species in order to better understand the unique features of the human auditory system that underlay speech perception.  The initial aim of the project will center around the organization of the auditory cortex of mice, bats, guinea pigs, ferrets, cats, macaques and humans.

==Research==
[[/Proposal|Research Proposal]]

[[/Progress|Progress Record]]

==Academic Background==

===Overview===
.
===Recommended Coursework===
[[Comparative Neuroscience]]

===Other Materials===
*[http://books.google.com/books?id=Mitx_JqHBNkC&amp;printsec=frontcover&amp;dq=The+symbolic+species&amp;source=bl&amp;ots=BsJ8_qm81n&amp;sig=TSili4bEYt20pKeYyo05fW1frIs&amp;hl=en&amp;ei=f4JPTNy6CoGC8gaoiK3bDQ&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=5&amp;ved=0CCkQ6AEwBA#v=onepage&amp;q&amp;f=false The Symbolic Species] by [[w:Terrence Deacon|Terrence Deacon]]
*[[Human Genetic Uniqueness Project]]

==References==
{{reflist}}

[[Category:Anatomy]]</text>
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    <title>Central Nervous System</title>
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      <text bytes="59556" sha1="gp1xo8uk0an2rxiend6wbzm2fiohkna" xml:space="preserve">{{biology}}
==== '''Introduction to the CNS''' ====

Most drugs that affect the central nervous system (CNS) act by altering some step in the neurotransmission process. Drugs may act presynaptically by influencing the production, storage, or termination of action of neurotransmitters. Other agents may activate or block postsynaptic receptors.&lt;br&gt;

In many ways, the basic functioning of neurons in the CNS is similar to that of the autonomic nervous system. E.g. transmission of information in the CNS and in the periphery both involve the release of neurotransmitters that diffuse across the synaptic space to bind to specific receptors on the postsynaptic neuron. In both systems, the recognition of the transmitter by the membrane receptor of the postsynaptic neuron triggers intracellular changes. Several major differences exists between neurons in the peripheral autonomic nervous system and those of the CNS. The circuitry of the CNS is much more complex than the autonomic nervous system, and the number of synapses in the CNS is far greater. The CNS, unlike the peripheral nervous system, contains powerful networks of inhibitory neurons that are constantly active in modulating the rate of neuronal transmission. In addition, the CNS communicates through the use of more than 50 different neurotransmitters. In contrast, the autonomic system uses only two primary neurotransmitters, acetylcholine and noradrenaline.&lt;br&gt;
== General Anesthetics ==

General anesthesia is essential to surgical practice because it renders patients&lt;br&gt;
(1) analgesics,&lt;br&gt;
(2) amnesic, &lt;br&gt;
(3) unconscious while causing &lt;br&gt;
(4) muscle relaxation and &lt;br&gt;
(5) suppression of undesirable reflexes.&lt;br&gt;
No single drug is capable of achieving these effects rapidly and safely. Rather, several different categories of drugs are utilized to produce “balanced anesthesia”. E.g. adjuncts to anesthesia consist of preanesthetic medication and skeletal muscle relaxants. Preanesthetic medication serves to calm the patient, relieve pain, and protect against undesirable effects of the subsequently administered anesthetic. Skeletal muscle relaxants facilitate intubation and suppress muscle tone to the degree required for surgery. Potent general anesthetics are delivered via inhalation or intravenous injection. With the exception of nitrous oxide, modern inhaled anesthetics are all volatile, halogenated hydrocarbons that derive early research and clinical experience with diethyl ether and chloroform. On the other hand, intravenous general anesthetics consist of a number of chemically unrelated drug types that are commonly used for the rapid induction of anesthesia.&lt;br&gt;

'''Concominant use of drugs'''&lt;br&gt;

Quite often, surgical patients receive one or more of the following preanesthetic medications: benzodiazepines (e.g. diazepam) to relieve anxiety and facilitate amnesia; barbiturates (e.g. pentobarbital) for sedation; antihistamines for prevention of allergic reactions (e.g. dimedrol); antiemetics (e.g. droperidol); opioids (e.g. fentanyl) for analgesia; anticholinergics (e.g. scopolamine) to prevent bradycardia and secretion of fluids into the respiratory tract. These agents facilitate smooth induction of anesthesia, and lower the dose of anesthetic required to maintain the desired level of surgical (Stage III) anesthesia.&lt;br&gt;

'''Induction, maintenance and recovery'''&lt;br&gt;

Anesthesia can be divided into three stages: induction, maintenance, and recovery. Induction is defined as the period of time from onset of administration of the anesthetic to the development of effective surgical anesthesia in the patient. Maintenance provides a sustained surgical anesthesia. Recovery is the time from discontinuation of administration of anesthesia until consciousness is regained. Induction of anesthesia depends on how fast effective concentration of anesthetic drug reach the brain; recovery is the reverse of induction and depends on how fast the anesthetic drug is removed from the brain.&lt;br&gt;

'''Depth of anesthesia'''&lt;br&gt;

The depth of anesthesia can be divided into a series of four sequential stages; each is characterized by increased CNS depression that is caused by accumulation of the anesthetic drug in the brain. With ether, which produce a slow onset of anesthesia, all the stages are discernible. However, with halothane and many other commonly used anesthetics, the stages are difficult to clearly characterize because of the rapidity of onset of anesthesia.&lt;br&gt;

'''Stage I–analgesia:''' Loss of pain sensation results from interference with sensory transmission in the spinothalamic tract. The patient is conscious and conversational. A reduced awareness of pain occurs as Stage II is approached.&lt;br&gt;

'''Stage II–excitement:''' The patient experiences delirium and violent combative behaviour. There is a rise and irregularity in blood pressure. The respiratory rate may be increased. To avoid this stage of anesthesia, a short acting barbiturate, such as sodium thiopental, is given i.v. before inhalation anesthesia is administered.&lt;br&gt;

'''Stage III'''–surgical anesthesia: Regular respiration and relaxation of the skeletal muscle occur in this stage. Eye reflexes decrease progressively, until the eye move-ments cease and the pupil is fixed. Surgery may proceed during this stage.&lt;br&gt;

'''Stage IV'''–medulary paralysis: Severe depression of the respiratory center and vasomotor center occur during this stage. Death can rapidly ensue.&lt;br&gt;

=== Inhalation Anesthetics ===
Inhaled gases are the mainstay of anesthesia and are primarily used for the maintenance of anesthesia after administration of an intravenous agent. Inhalation anesthetics have a benefit that is not available with intravenous agents, since the depth of anesthesia can be rapidly altered by changing the concentration of the inhaled anesthetic. Because most of these agents are rapidly eliminated from the body, they do not cause postoperative respiratory depression.&lt;br&gt;

A. Common features of inhaled anesthetics: Modern inhalation anesthetics are non-explosive agents that include the gas nitrous oxide as well as a number of volatile halogenated hydrocarbons. As a group, these agents decrease cerebrovascular resistance, resulting in increased perfusion of the brain. They cause bronchodilation and decrease minute ventilation. Their clinical potency cannot be predicted by their chemical structure, but potency does correlate with their solubility in lipid. The movement of these agents from the lungs to the different body compartments depends upon their solubility in blood and various tissues. Recovery from their effects is due to their redistribution from the brain.&lt;br&gt;

B. Potency: The potency of inhaled anesthetics is defined quantitatively as the minimum alveolar concentration (MAC), which is the concentration of anesthetic gas needed to eliminate movement among 50% of patients challenged by standardized skin incision. The MAC is usually expressed as the percent of gas in a mixture required to achieve the effect. Numerically, is small for potent anesthetics, such as halothane, and large for less potent agents, such as nitrous oxide (see diagram 6.1). The MAC values are useful in comparing pharmacologic effects of different anesthetics. The more lipid-soluble an anesthetic, the lower the concentration of anesthetic needed to produce anesthesia.&lt;br&gt;

. Uptake and Distribution: The partial of an anesthetic gas at the origin of the respiratory pathway is the driving force that moves the anesthetic into the alveolar space and thence into the blood, which delivers the drug to the brain and various other body compartments. Since gases move from one compartment to another within the body according to partial pressure gradients, a steady state is achieved when the partial pressure in each of these compartments is equivalent to that in the inspired mixture. The time course for attaining this steady state is determined by the following three factors:&lt;br&gt;

1. alveolar wash-in: This term refers to the replacement of the normal lung gases with the inspired anesthetic mixture. The time required for this process is directly proportional to the functional residual capacity of the lung, and inversely proportional to the ventilatory rate; it is independent to the physical properties of the gas. Once the partial pressure builds within the lung, anesthetic uptake from the lung begins.&lt;br&gt;

2. solubility in the blood: The first compartment that the anesthetic gas encountersis the blood. Solubility  in blood is determined by a physical property of the anesthetic molecule called the blood/gas partion coefficient, which is the ratio of the total amount of gas in the blood relative to the gas equilibrium phase. Drugs with low versus high solubility in blood differ in their speed of induction of anesthesia. E.g. when an anesthetic gas with low blood solubility, such as nitrous oxide, diffuses from the alveoli into the circulation, little of the anesthetic dissolves in the blood. Therefore, the equilibrium between the inhaled anesthetic and arterial blood occur rapidly, and relative few additional molecules of anesthetic are required to raise arterial tension (that is, steady state is rapidly achieved). In contrast, an anesthetic gas with high blood solubility, such as halothane, dissolves more completely in the blood, and greater amount of the anesthetic and longer periods of time are required to raise arterial tension. This results in increased time of induction and recovery, and slower changes in the depth of anesthesia in response to changes in the concentration of the inhaled drug.&lt;br&gt;

3. tissue uptake: The arterial circulation distributes the anesthetic to various tissues, and the pressure gradient drives free anesthetic gas into tissues. The time required for a particular tissue to achieve a steady-state is inversely proportional to the blood flow to that tissue (faster flow results in a more rapidly achieved steady-state), and directly proportional to the capacity to store anesthetic (larger capacity results in a longer time to achieve steady-state). Capacity, in turn, is directly proportional to the tissue’s volume, and the tissue/blood solubility coefficient of the anesthetic. On the basis of these considerations, four major compartments determine the time course of anesthetic uptake:&lt;br&gt;

a.	Brain, heart, liver, kidney, endocrine glands: These highly perfused tissues rapidly attain a steady-state with the partial pressure of the anesthetic in blood.&lt;br&gt;

b.	Skeletal muscles: These are poorly perfused during anesthesia. This fact prolongs the time required to achieve steady-state.&lt;br&gt;

c.	Fat: This tissue is also poorly perfused. However, potent general anesthetics are very lipid soluble. Therefore, fat has a large capacity to store anesthetic. This combination of slow delivery to a high capacity prolongs the time required to achieve steady state.&lt;br&gt;

d.	Bone, ligaments, and cartilage: These are poorly perfused and have a relatively low capacity to store anesthetic. Therefore, these tissues have only a slight impact on the time course of anesthetic distribution in the body.&lt;br&gt;

4. washout: When the administration of an inhalation anesthetic is discontinued, the body now becomes the “source” that drives the anesthetic into the alveolar spece. The same factors that influence attainment of steady-state with an inspired anesthetic determine the time course of clearace of the drug from the body.&lt;br&gt;

D. Specific inhalation anesthetics
Each of the halogenated gases has characteristics beneficial for selected clinical applications. No one anesthetic is superior to another under all circumstances. [Note: In a very small population of patients, all of the halogenated hydrocarbon anesthetics have the potential to induce malignant hyperthermia. While the etiology of this condition is unknown, it appears to be inherited. Should a patient exhibit the hyperthermia and muscle rigidity characteristic to malignant hyperthermia, dantrolene is given as the anesthetic mixture is withdrawn.]&lt;br&gt;

1. Halothane (ftorothane): This agent is the prototype to which newer agents in this series of anesthetics are compared. While halothane is a potent anesthetic, it is a relatively weak analgesic. Thus, halothane is usually co-administered with nitrous oxide, opioids,or local anesthetics. Like other halogenated hydrocarbons, halothane is vagomimetic and will cause atropine-sensitive bradycardia. In addition, halothane has the undesirable property of causing cardiac arrhythmias. Halothane is oxidatively metabolized in the body to tissue-toxic hydrocarbons (e.g. trifluroethanol) and bromide ion. These substances may be responsible for the toxic reactions that same patients (especially females) develop after halothane anesthesia. This reaction begins as a fever, anorexia, nausea, and vomiting, and patients may exhibit signs of hepatitis. Although the incidence of this reaction is low––approximately 1 in 10,000 individuals––50% of such patients will die of hepatic necrosis. [Note: Halothane is not hepatotoxic in pediatric patients and that, combined with its pleasant odor, make it the agent of choice in children.] Halothane anesthesia is not repeated at intervals less than 2 to 3 weeks.&lt;br&gt;

2. Enflurane: This gas is less potent than halothane but it produces rapid induction and recovery. About 2% of the agent is metabolized to fluoride ion, which is excreted by the kidney. Therefore, enflurane is contraindicated in patients with kidney failure. Enflurane anesthesia exhibits the following differences from halothane: fewer arrhythmias, less sensitization of the heart catecholamines, and greater potentiation of muscle relaxants. A disadvantage of enflurane is that it causes CNS excitation at twice the MAC and at lower doses if hyperventilation reduces the pCO2.&lt;br&gt;

3. Isoflurane: This is a newer halogenated anesthetic that has low biotransformation and low organ toxicity. Unlike the other halogenated anesthetic gases, isoflurane does not induce cardiac arrhythmias and does not sensitize the heart to the action of catecholamines. Isoflurane is a vary stable molecule that undergoes little metabolism, as a result of which, less fluoride is produced. Isoflurane is not currently believed to be tissue toxic.&lt;br&gt;

4. Methoxyflurane: The agent is the most potent inhalation anesthetic because of its high solubility in lipid. Prolonged administration of methoxyflurane is associated with metabolic release of fluoride, which is toxic to the kidneys. Therefore, methoxyflurane is rarely used outside obstetric practice. It finds use in child-birth because it does not relax the uterus when briefly inhaled.&lt;br&gt;

5. Nitrous oxide: Whereas nitrous oxide (N2O or “laughing gas”) is a potent analgesic, it is a weak general anesthetic. Thus, it is frequently combined with other more potent agents. Because it moves very rapidly into and out of the body, nitrous oxide can increase the volume (pneumothorax) or pressure (sinuses) within closed body compartments. Furthermore, its speed of movement allows nitrous oxide to retard oxygen uptake during recovery, thus causing diffusion hypoxia. This anesthetic does not depress respiration nor does it produce muscle relaxation. It also has the least effect on cardiovasculaar system and on increasing cerebral blood flow, and is the least hepatotoxic of the inhalation anesthetics. It is therefore probably the safest of these anesthetics, provided that at least 20% of oxygen is always administered at the same time. [Note: Nitrous oxide at 80% (without other adjunct agents) cannot produce surgical anesthesia.] Nitrous oxide is often employed of 30% in combination with oxygen for analgesia, particularly in dental surgery.&lt;br&gt;

=== Intravenous Anesthetics===
Intravenous anesthetics are often used for the rapid induction of anesthesia, which is then maintained with an appropriate inhalation agent. They rapidly induce anesthesia, and must therefore be injected slowly. Recovery from intravenous anesthetics is due to redistribution from sites in the CNS.&lt;br&gt;

'''A. Barbiturates'''&lt;br&gt;

Thiopental is a potent anesthetic and a weak analgesic. It is the most widely used intravenously administered general anesthetic. It is an ultra-short-acting barbiturate and has a high lipid solubility. When thiopental is administered intravenously, it quickly enters the CNS and depresses function, often in less than 1 minute. However, diffusion out of the brain can occur very rapidly as well, because of redistribution of the drug to other body tissues, including skeletal muscle and ultimately adipose tissue. This latter site serves as a reservoir of drug from which the agent slowly leaks out and is metabolized and excreted. The short duration of action is due to the decrease of its concentration in the brain. Thus, metabolism of thiopental is much slower than tissue redistribution. The barbiturates are not significantly analgesic and require some type of supplementary analgesic administration during anesthesia. Thiopental has minor effects on the cardiovascular system, but it may contribute to severe hypotension in hypovolemic or shock patients. All barbiturates can cause apnea, coughing, chest wall spasm, laryngospasm, and bronchospasm. Barbiturates are contraindicated in patients with acute intermittent or variegate porphyria.&lt;br&gt;

'''B. Benzodiazepines'''&lt;br&gt;

Although diazepam is the prototype benzodiazepine, lorazepam and midazolam are more potent. All three facilitate amnesia while causing sedation. Midazolam has become popular because of its combination of water solubility, rapid onset and short duration of action. It produces amnesia with few side effects. Mental function returns to normal within 4 hours, making it popular choice for ambulatory surgery and during regional anesthesia.&lt;br&gt;

'''C. Opioids'''&lt;br&gt;

	Because of their analgesic property, opioids are frequently used together with other anesthetics; e.g. the combination of morphine and nitrous oxide provide good anesthesia for cardiac surgery. However, opioids are not good amnesics and they can cause hypotension, respiratory depression, and muscle rigidity as well as post-anesthetic nausea and vomiting. Fentanyl is more frequently used than morphine. &lt;br&gt;

	The combination of droperidol and fentanyl is a fixed ration preparation called thalamonal (innovar). Since droperidol is a neuroleptic substance, THALAMONAL is said to produce neurolept analgesia. A neuroleptic has adrenergic blocking as well as sedative, antiemetic properties.&lt;br&gt;

'''D. Ketamine'''&lt;br&gt;

	Ketamine, a short-acting nonbarbiturate anesthetic, induces a dissociated state in which the patient appears awake but is unconscious and does not feel pain. This dissociative anesthesia provides sedation, amnesia, and immobility. Ketamine stimulates the central sympathetic outflow, which in turn, causes stimulation of the heart and increases blood pressure and cardiac output. Ketamine is therefore used when circulatory depression is undesirable. On the other hand, these effects mitigate against the use of ketamine in hypertensive or stroke patients. The drug is lipophilic and enters the brain very quickly, but like the barbiturates, it can redistribute to other organs and tissues. It is metabolized in the liver. Ketamine is employed mainly in children and young adults for short procedures. It is not widely used because it increases cerebral blood flow and induces postoperative hallucinations (nightmares).&lt;br&gt;

'''E. Propanidide'''&lt;br&gt;

	Propanidide (Sombrevin) is an ultra-short-acting intravenous anesthetic. Onset is smooth and occurs within about 40 seconds of administration. The emergence from anesthesia from propanidide is more rapid than that from thiopental and is characterized by minimal posoperative confusion.
'''F. Sodium oxybutyrate''' &lt;br&gt;

Sodium oxybutirate is a derivative of -aminobutyric acid (GABA). GABA is an inhibitory neurotransmitter in the CNS. In contrast to GABA, sodium oxybutyrate readily crosses the blood-brain barier and produces sedative, hypnotic, anesthetic, anticonvulsive, and antihypoxic effects. However, sodium oxybutyrate is not good analgesic. The onset of action is slow (in 30-40 minutes after i.v. injection). It must be used with other anesthetics for surgical anesthesia.&lt;br&gt;

==Ethyl Alcohol (Ethanol)==
Pharmacology of alcohol is important for its presence in beverages (which have been used since recorded history) and for alcohol intoxication, rather than as a drug. Alcohol is manufactured by fermentation of sugar:&lt;br&gt;

C6H12O6                        2CO2 + 2C2H5OH&lt;br&gt;

Fermentation proceeds till alcohol content reaches 15%. Then reaction is inhibited by alcohol itself.&lt;br&gt;

'''1. Local action:''' Rubbed on skin ethanol dissolves fat, precipitate surface proteins and harden skin. By precipitation bacterial proteins it acts as an antiseptic. The antiseptic action increases with concentration from 20% to 70%, remain constant from 70 to 90% and decreases above that. 100% ethanol is more dehydrating but poorer antiseptic than 90% ethanol. Applied to delicate skin or mucous membrane ethanol produces irritation and burning sensation. Injected s.c. it causes intense pain, inflammation and necrosis followed by fibrosis. Injected round a nerve it produces permanent damage.&lt;br&gt;

'''2. Resorbtive action:''' Ethanol is a neuronal depressant. However, as highest areas are most sensitive to alcohol (these are primarily inhibitory),––excitation and euphoria are experienced at lower plasma concentrations (30-100mg/dl). Caution, self criticism and restraint are lost first. With the increasing concentration (100-150mg/dl) mental clouding, disorganization of thought and drowsiness occur. At 50-200mg/dl the person is sloppy, ataxic and drunk; 200-300mg/dl results in stupor and above this unconsciousness prevails, medulary centers are paralyzed and death may occur. Though, ethanol can produce anesthesia, margin of safety is narrow. Effects of alcohol are more marked when the concentration is rising than when it is falling.&lt;br&gt;

Small doses of alcohol produce cutaneous and gastric vasodilation, a sense of warmth, but heat lost is increased in cold surroundings.
Alcoholic beverages have variable effect on gastric secretion depending on beverage itself and whether the individual likes it. Dilute alcohol (optimum 10%) is a strong stimulant of gastric secretion. Higher concentrations (above 20%) inhibit gastric secretion.
Regular intake of small amounts of alcohol has been found to raise high density lipids and diminish low density lipids levels in plasma. This may be responsible for the somewhat lower incidence of coronary artery disease.
'''3. Pharmacokinetics:''' Rate of alcohol absorption from stomach is dependant on its concentration, presence of food, and other factors. Absorption from intestines is very fast. Thus, gastric emptying determines rate of absorption. Alcohol distributes widely in the body, crosses blood-brain barrier (concentration in the brain is very near to blood concentration); it also crosses placenta freely. It is oxidized in liver.&lt;br&gt;

Metabolism of alcohol follows zero order kinetics, i.e. a constant amount is degraded in unit time, irrespective to blood concentration (8-10 ml of absolute alcohol/hour). Small amounts are excreted through kidney and lungs.&lt;br&gt;

'''4. Acute alcoholic intoxication:''' Nausea, vomiting, hypotension, hypoglycemia, collapse, respiratory depression, coma and death.
'''Treatment:''' Gastric lavage, maintain patent airway and aspiration of vomitus. Most patient will recover with supportive treatment and maintenance of fluid and electrolyte balance. Insuline+fructose has been found to accelerate alcohol metabolism.&lt;br&gt;

Disulfiram (teturam, esperal) has found some use in the patients seriously desiring to stop alcohol ingestion. The drug blocks the oxidation of acetaldehyde to acetic acid by inhibiting aldehyde dehydrogenase. This result in accumulation of acetaldehyde in the blood, causing flushing, tachycardia, hyperventilation, and nausea.&lt;br&gt;

 
==Hypnotic Drugs ==

Sleep is an active, circadian, physiological depression of consciousness. It is characterized by cyclical electroencephalographic (EEG) and eye movement changes. Normal sleep (categorized by eye movement) is of two kinds:&lt;br&gt;

1. NREM (non-rapid eye movement), orthodox, forebrain or slow-wave EEG sleep. Heart rate, blood pressure and respiration are steady or decline and muscles are relaxed; growth hormone secretion is maximal.&lt;br&gt;

2. REM (rapid eye movement), paradoxical, hindbrain of fast-wave EEG sleep; awakened subjects state they were ‘dreaming’. Heart rate, blood pressure and respiration are increased, skeletal muscles are profoundly relaxed.&lt;br&gt;

Cycles of NREM sleep (about 90 min) alternate with REM sleep (about 20 min). thus, normal sleep contains 4-6 cycles (NREM+REM).
Classification of insomnia (sleep disorders):&lt;br&gt;

a.Difficulty in getting to sleep, onset insomnia; this was associated with neurotic personality disorder. People who lie awake in bed for hours unable to relax, and then sleep well.&lt;br&gt;

b.Difficulty in staying asleep, repeated awakenings.&lt;br&gt;

c.Early waking in which sleep is shorter.&lt;br&gt;

In general, a short-acting hypnotic drug  is prescribed for patients who have prolonged sleep latency but who sleep well once sleep ensues; and a drug with a longer duration of action for those who awaken early and have difficulty in returning to sleep. Insomnia has many causes, and an accurate differential diagnosis is required before treatment should be considered. Prescription of a hypnotic without regard to the underlying disturbance subjects the patient to the risk of abuse, may mask the signs and symptoms of a pernicious disease etc.&lt;br&gt;

=== Barbituates ===

Barbital was introduced in 1903 and phenobarbital in 1912. The barbiturates were formerly the mainstay of treatment used to sedate the patient or to induce and maintain sleep. Today, they have been largely replaced by the benzodiazepines, mainly because barbiturates induce tolerance, drug-metabolizing enzymes, physical dependence, and very severe withdrawal symptoms. Foremost is their ability to cause coma in toxic doses. Certain barbiturates, such as the very short-acting thiopental, are still used to induce anesthesia.&lt;br&gt;

'''1. Mode of action'''&lt;br&gt;

Barbiturates are thought to interfere with Na+ and K+ transport across cell membrane. This leads to inhibition of the mesencephalic reticular activation system. Polysynaptic transmission is inhibited in all areas of the CNS. Barbiturates also potentiate GABA action on chloride entry into the neuron, although they do not bind at the benzodiazepine receptor.&lt;br&gt;

'''2.Actions'''&lt;br&gt;

2.1 Depression of the CNS: The barbiturates can produce all degrees of depression of the CNS. Low doses produce sedation (calming effect, reducing excitement). At high doses, the drugs cause hypnosis, followed by anesthesia (loss of feeling or sensation), and finally coma and death. Thus, any degree of depression of the CNS is possible, depending on the dose. Barbiturates do not raise the pain threshold and have no analgesic properties. They may even exacerbate pain.&lt;br&gt;

2.2 Respiratory depression: Barbiturates suppress the hypoxic and chemoreceptor response to CO2, and overdosage is followed by respiratory depression and death.&lt;br&gt;

2.3 Enzyme induction: Barbiturates induce P-450 microsomal enzymes in the liver. Therefore, chronic barbiturate administration diminishes the action of many drugs that are dependant on P-450 metabolism (including barbiturates itself).&lt;br&gt;

Barbiturates are classified according to their duration of action:&lt;br&gt;

1. Long-acting (phenobarbital, barbital, barbital-sodium), which have a duration of action greater than a day; phenobarbital is useful in the treatment of seizures.&lt;br&gt;

2. Intermediate-acting (pentobarbital, cyclobarbital, secobarbital), which are effective as sedative and hypnotic (but not antianxiety) agents.&lt;br&gt;

3. Short-acting (hexobarbital).&lt;br&gt;

4. Ultra short-acting (thiopental, hexenal), which act within seconds and have a duration of action of about 30 minutes, are used in the intravenous induction of anesthesia.&lt;br&gt;

'''3. Therapeutic uses'''&lt;br&gt;

3.1 Anesthesia: Selection of a barbiturate is strongly influenced by the desired duration of action. The ultra-short-acting barbiturates, such as thiopental, are used intravenously to induce anesthesia.&lt;br&gt;

3.2 Anticonvulsant: Phenobarbital is used in long-term management of tonic-clonic seizures, status epilepticus, and eclampsia. Phenobarbital has been regarded as the drug of choice for treatment of young children with recurrent febrile seizures. However, phenobarbital can depress cognitive performance in children, and the drug should be used cautiously. Phenobarbital has specific anticonvulsant activity that is distinguished from the nonspecific CNS depression.&lt;br&gt;

3.3 Sedation-hypnosis: Barbiturates have been used as mild sedative to relieve nervous tension, and insomnia. Most have been replaced by benzodiazepines.&lt;br&gt;

'''4. Pharmacokinetics'''&lt;br&gt;

Barbiturates are absorbed orally and distributed widely throughout the body. All barbiturates redistribute from the brain to the splanchnic areas, to skeletal muscle, adipose tissue. This movement is important in causing the short duration of action of thiopental. Barbiturates are metabolized in the liver, and inactive metabolites are excreted in the urine.&lt;br&gt;

'''5. Adverse effects'''&lt;br&gt;

5.1 CNS: Barbiturates cause drowsiness, impaired concentration, and mental and physical slugginess.&lt;br&gt;

5.2 Drug hangover: Hypnotic doses of barbiturates produces feeling of tiredness well after the patient awakes. This drug hangover leads to impaired ability to function normally for many hours after waking.&lt;br&gt;
 
5.3 Addiction: Abrupt withdrawal from barbiturates may cause tremors, anxiety, weakness, restlessness, nausea and vomiting, seizures, delirium, and cardiac arrest. Withdrawal is much more severe than that associated with opiates and can result in death.&lt;br&gt;

5.4 Precautions: As noted previously, barbiturates induce the P-450 system and there-fore may decrease the effect of drugs that are metabolized by these hepatic enzymes. Barbiturates increase porphyrin synthesis, and are contraindicated in patients with acute intermittent porphyria.&lt;br&gt;

'''6. Poisoning'''&lt;br&gt;

Barbiturate poisoning has been a leading cause of death among drug overdoses for many decades. Severe depression of respiration is coupled with central cardiovascular depression, and results in a shock-like condition with shallow, infrequent breathing. Treatment includes artificial respiration and purging the stomach of its contents if the drug has been recently taken. Hemodialysis may be necessary if large quantities have been taken. Alkalinization of the urine often aids in the elimination of phenobarbital. Analeptics (bemegrid, corazole) may be useful if respiration is not depressed completely to restore breathing.&lt;br&gt;

===Benzodiazepines ===

Chlordiazepoxide was synthesized in 1957 by Sternbach. Randall discovered its unique pattern of actions. With its the introduction into clinical medicine in 1961, more than 3000 benzodiazepines have been synthesized, over 120 have been tested for biological activity, and about 35 are in clinical use in various parts of the world. They have largely replaced barbiturates and meprobamate as sedative-hypnotic agents mainly because of remarkably low capacity to produce fatal CNS depression.&lt;br&gt;

Although the benzodiazepines exert qualitatively similar effects, important quantitative differences in their pharmacodinamic spectra and pharmacokinetic properties have led to varying patterns of therapeutic application. Benzodiazepines vary in the degrees of sedative-hypnotic, muscle relaxant, anxiolytic, and anticonvulsant effects. &lt;br&gt;

'''Mode of action'''&lt;br&gt;

Binding of -aminobutyric acid (GABA) to its receptor on the cell membrane triggers an opening of a chloride channel, which leads to an increase in chloride conductance.&lt;br&gt;

The influx of chloride ions causes a small hyperpolarization that moves the postsynaptic potential away from its firing threshold and thus inhibits the formation of action potentials. Benzodiazepines bind to specific, high affinity sites on the cell membrane, which are separate from but adjacent to the receptor of GABA. The benzodiazepine receptors are found only in the CNS, and their location parallels that of GABA neurons. The binding of benzodiazepines enhances the affinity of GABA receptors for its neurotransmitter, resulting in a more frequent opening of adjacent chloride channels. This in turn results in enhanced hyperpolarization and further inhibition of neuronal firing.&lt;br&gt;

'''Actions'''&lt;br&gt;

The benzodiazepines are not general neuronal depressants, as are the barbiturates. All the benzodiazepines have quite similar pharmacological profiles. The most prominent of these effects are sedation, hypnosis, decreased anxiety, muscle relaxation, anterograde amnesia, and anticonvulsant activity. Nevertheless, the drugs differ in selectivity, and the clinical usefulness of individual benzodiazepines varies considerably.&lt;br&gt;

'''Uses in sleep disorders'''&lt;br&gt;

This lecture describes the benzodiazepines used primarily as hypnotic and anti-convulsant agents. Other applications will be discussed in details in a following lecture (see tranquilizers). Not all of the benzodiazepines are useful as hypnotic agents, although all have sedative or calming effects. The three most commonly prescribed benzodiazepines for sleep disorders are long-acting flurazepam, intermediate-acting temazepam, and short-acting triazolam. The principal factors that determine selection are pharmacokinetics.&lt;br&gt;

'''a. Flurazepam:''' This long-acting benzodiazepine significantly reduces both sleep-induction time and the number of awakening, and increases the duration of sleep. With continued use, the drug has been shown to maintain its effectiveness for up to 4 weeks. Flurazepam has a half-life of approximately 85 hours, which may result in daytime sedation and accumulation of drug.&lt;br&gt;

'''b. Temazepam:''' this drug is useful in patients who experience frequent awakening. However, the peak sedative effect occurs two to three hours after an oral dose, and therefore it may be given several hours before bedtime.&lt;br&gt;

'''c. Triazolam:''' This benzodiazepine has a relatively short duration of action and is therefore used to induce sleep in patients with recurring insomnia. Whereas temazepam is useful for insomnia caused by the inability to stay asleep, triazolam is effective in treating individuals who have difficulty in going to sleep. Tolerance frequently develops within a few days, and withdrawal of the drug often results in rebound insomnia, leading the patient to demand another prescription. Therefore, this drug is best used intermittently rather than daily. In general, hypnotics should be given for only a limited time, usually less than 2 to 4 weeks.&lt;br&gt;

'''Pharmacokinetics'''&lt;br&gt;

1. Absorption and distribution: The benzodiazepines are lipophilic and are rapidly and completely absorbed after oral administration and are distributed throughout the body.&lt;br&gt;

2. Duration of action: The half-lives of the benzodiazepines are very important clinically, since the duration of action may determine the therapeutic usefulness. The benzodiazepines can be roughly divided into short-, intermediate- and long-acting groups. The long-acting agents form active metabolites with long half-lives.&lt;br&gt;

3. Fate: Most benzodiazepines, including chlordiazepoxide and diazepam, are metabolized by the hepatic microsomal metabolizing system to compounds that are also active. For these benzodiazepines, the apparent half-life of the drug represents the combined actions of the parent drug and its metabolites. The benzodiazepines are excreted in urine as glucuronides or oxidized metabolites.&lt;br&gt;

'''Dependence'''&lt;br&gt;

Psychological and physical dependence on benzodiazepines can develop if high doses of the drug are given over a prolonged period. Abrupt discontinuation of the benzo-diazepines results in withdrawal symptoms, including confusion, anxiety, agitation, restlessness, insomnia, and tension. Because of the long half-lives of some of the benzodiazepines, withdrawal symptoms may not occur until a number of days after discontinuation. Benzodiazepines with a short elimination half-life, such as triazolam, induce more abrupt and severe withdrawal reactions.&lt;br&gt;

'''Adverse effects'''&lt;br&gt;

Drowsiness and confusion are the two most common side effect of the benzo-diazepines. Ataxia occurs at high doses and precludes activities that require motor coordination, such as driving an automobile. Cognitive impairment (decreased long-term recall and acquisition of new knowledge) can occur. Triazolam, the benzodiazepine with most rapid elimination, often show a rapid development of tolerance, early morning insomnia and daytime anxiety, along with amnesia and confusion. Benzodiazepines potentiate alcohol and other CNS depressants. However, they are considerably less dangerous than other anxiolytic and hypnotic drugs. As a result, a drug overdose is seldom lethal, unless other central depressants, such as alcohol, are taken concurrently.&lt;br&gt;

Benzodiazepine antagonist flumazenil is a GABA receptor antagonist that can rapidly reverse the effects of benzodiazepines. The drug is available by i.v. administration only. Onset is rapid but duration is short, with a half-life of about one hour. Frequent administration may be necessary to maintain reversal of benzodiazepines.&lt;br&gt;

=== Other Hypnotic Agents ===

Although the hypnotic zopiclone (imovan) is not a benzodiazepine, it acts on a subset of the benzodiazepine receptor family. Zopiclone has no anticonvulsive or muscle relaxing properties. It shows no withdrawal effects, exhibits minimal rebound insomnia and little or no tolerance occurs with prolonged use. Zopiclone has a rapid onset of action and short elimination. Although zopiclone potentially has advantage over the benzodiazepines, clinical experience with the drug is still limited.&lt;br&gt;

Chloral hydrate is a trichlorinated derivative of acetaldehyde that is converted to trichlorethanol in the body. The drug is an effective sedative and hypnotic that induces sleep in about 30 minutes and lasts about 6 hours. Chloral hydrate is irritating to the gastrointestinal tract and causes epigastric distress. It also produces an unusual, unpleasant taste sensation.&lt;br&gt;

== Anticonvulsants ==

=== Antiepileptic Drugs ===
Epilepsy is widespread among the general population. Epilepsy is not a single entity; it is a family of different recurrent seizure disorders that have in common the sudden and excessive discharge of cerebral neurons. This results in abnormal movements or perceptions that are of short duration but that tend to recur. The site of electrical discharge determines the symptoms that are produced. E.g. epileptic seizures may cause convulsions if the motor cortex is involved. The seizures may include visual, auditory, or olfactory hallucinations if the parietal or occipital cortex plays a role. Drug therapy is the most widely effective mode of treatment for epilepsy. Seizures can be controlled completely in approximately 50% of epileptic patients, and meaningful improvement may be achieved in at least one half of the remaining patients.&lt;br&gt;

'''A. Etiology'''&lt;br&gt;

The neuronal discharge in epilepsy results from the firing of a small population of neurons in some specific area of the brain, referred to as the primary focus. These focal areas may be triggered into activity by changes in any of a variety of environmental factors, including alteration in blood gases, pH, electrolytes, or glucose availability.&lt;br&gt;

1.Primary epilepsy: When no specific anatomic cause for the seizure, such as trauma or neoplasm, is evident the syndrome is called primary epilepsy. These seizures may be produced by an inherited abnormality in the CNS. Patients are treated chronically with antiepileptic drugs, often for life.&lt;br&gt;

2.Secondary epilepsy: A number of reversible disturbances, such as tumors, head injury, hypoglycemia, meningeal infection, or rapid withdrawal of alcohol can precipitate seizures. Antiepileptic drugs are given until the primary cause can be corrected.&lt;br&gt;
 
'''B. Classification of epilepsy'''&lt;br&gt;

Seizures have been classified into two broad groups, partial (or focal), and generalized. Choice of drug treatment is based on the classification.&lt;br&gt;

'''1. Partial:''' The symptoms of each seizure type depend on the site of neuronal discharge and on the extent to which the electrical activity spreads to other neurons in brain. Partial seizures may progress, becoming generalized tonic- clonic seizures.&lt;br&gt;

1.1 Simple partial: These seizures are caused by a group of hyperactive neurons and are confined to a single locus in the brain; the abnormal activity does not spread. The patient does not lose consciousness and often exhibits abnormal activity of a single limb or muscle group that is controlled by the region the brain experiencing the disturbance. The patient may also show sensory distortions. Simple partial seizures may occur at any age.&lt;br&gt;

1.2 Complex partial: These seizures exhibit complex sensory hallucinations, mental distortion, and loss of consciousness. Motor dysfunction may involve chewing movements, diarrhea, urination. Most (80%) of individuals experience their initial seizures before 20 years of age.&lt;br&gt;

'''2. Generalized:''' These seizures begin locally, but they rapidly spread, producing abnormal electrical discharge throughout both hemispheres of the brain. Generalized seizures may be convulsive or nonconvulsive; the patient usually has an immediate loss of consciousness.&lt;br&gt;

2.1 Tonic-clonic (grand mal): This is the most commonly encountered and the most dramatic form of epilepsy. Seizures result in loss of consciousness, followed by tonic, then clonic phases. The seizure is followed by a postictal period of confusion and exhaustion.&lt;br&gt;

2.2 Absence (petit mal): These seizures involve a brief, abrupt, and self-limiting loss of consciousness. The onset occurs in patients at age 3 to 5 years and lasts until puberty. The patient stares and exhibits rapid eye-blinking, which lasts for 3 to 5 seconds.&lt;br&gt;

2.3 Myoclonic: These seizures consist of short episodes of muscle contractions that may reoccur for several minutes. Myoclonic seizures are rare, occur at any age, and are often a result of hypoxia, uremia, encephalitis, or drug poisoning.&lt;br&gt;

2.4 Febrile seizures: Young children (3 month to 5 years of age) frequently develop seizures with illness accompanied by high fever. The febrile consist of generalized tonic-clonic convulsions of short duration. Although febrile seizures may be frighten-ing, they are benign and do not cause death, neurologic damage, injury, or learning disorders, and they rarely require medication.&lt;br&gt;

2.5 Status epilepticus: Seizures are rapidly recurrent.&lt;br&gt;

'''3. Mechanism of action of antiepileptic drugs'''&lt;br&gt;

Drugs that are effective in seizure reduction can either block the initiation of the electrical discharge from the focal area or, more commonly, prevent the spread of the abnormal electrical discharge to adjacent brain areas.&lt;br&gt;

Initial drug treatment is based on the specific type of seizure. Thus, tonic-clonic (grand mal) seizures are treated differently than absence (petit mal). Several drugs may be equally effective, and the toxicity of the agent is often a major consideration in drug selection.
Diphenin (phenytoin) is effective in suppressing tonic-clonic and partial seizures, and is a drug of choice for initial therapy, particularly in treating adults.&lt;br&gt;

1. Mechanism of action: Diphenin stabilizes neuronal membranes to depolarization by decreasing the flux of sodium ions in neurons in the resting state or during depolarization and suppresses repetitive firing of neurons.&lt;br&gt;

2. Actions: Diphenin is not a generalized CNS depressant like the barbiturates, but it does produce some degree of drowsiness and lethargy. Diphenin reduces the propagation of abnormal impulses in the brain.
3. Therapeutic uses: Diphenin is highly effective for all partial seizures (simple and complex), for tonic-clonic seizures, and in status epilepticus. Diphenin is not effective for absence seizures, which often may worsen if such a patient is treated with this drug.
Carbamazepine
1. Action: Carbamazepine reduces the propagation of abnormal impulses in the brain by blocking sodium channels, thereby inhibiting the generation of repetitive action potentials in the epileptic focus.&lt;br&gt;

2. Therapeutic uses: Carbamazepine is highly effective for all partial seizures and is often the drug of first choice. In addition the drug is highly effective for tonic-clonic seizures and is used to treat trigeminal neuralgia. It has occasionally been used in manic-depressive patients to ameliorate the symptoms.&lt;br&gt;

Phenobarbital provides a 50% favorable response rate for simple partial seizures, but it is not very effective for complex partial seizures. The drug has been regarded as the first choice in treating recurrent seizures in children, including febrile seizures. The drug is also used to treat tonic-clonic seizures, especially in patients who do not respond to diazepam plus diphenin. Doses required for antiepileptic action are lower than those that cause pronounced CNS depression. Phenobarbital is also used as a mild sedative to relieve anxiety, nervous tension and insomnia, although benzodiazepines are superior.&lt;br&gt;

Hexamidin (Primidone) is structurally related to phenobarbital, and resembles phenobarbital in its anticonvulsive activity. Hexamidin is an alternative choice in partial seizures and tonic-clonic seizures. Much of hexamidin’s efficacy comes from its metabolites phenobarbital and phenylethylmalonamide.&lt;br&gt;

Valproic acid is the most effective agent available for treatment myoclonic seizures. The drug diminishes absence seizures but is a second choice because of its hepatotoxic potential. Valproic acid also reduces the incidence and severity of tonic-clonic seizures.&lt;br&gt;

Ethosuximide reduces propagation of abnormal electrical activity in the brain, and is the first choice in absence seizures.&lt;br&gt;

Several of the benzodiazepines show antiepileptic activity. Clonazepam and clorazepate are used for chronic treatment, whereas diazepam is the drug of choice in the acute treatment of status epilepticus. Of all the antiepileptics, the benzodiazepines are the safest.&lt;br&gt;

=== Drugs Used in Parkinson's Disease ===
Parkinsonism is a progressive neurologic disorder of muscle movement, characterized by tremors, muscular rigidity, and bradykinesia (slowness in initiating and carrying out voluntary movements). Parkinson’s disease is the fourth most common neurologic disorder among the elderly. Most cases involve people over the age of 65 among whom the incidence is about 1:100 individuals.&lt;br&gt;

'''A. Etiology'''&lt;br&gt;

The cause of Parkinson’s disease is unknown for most patients. The disease is correlated with a reduction in the activity of inhibitory dopaminergic neurons in the substantia nigra and corpus striatum–parts of the brain’s basal ganglia system that are responsible for motor control. Genetic factors do not play a dominant role in the etiology.&lt;br&gt;

1. Substantia nigra: The substantia nigra, part of the extrapyramidal system, is the source of dopaminergic neurons that terminate in the striatum. Each neuron makes thousands synaptic contacts within the striatum and modulates the activity of a large number of cells. These dopaminergic projections from the substantia nigra fire tonically, rather than in response specific muscular movements or sensory input. Thus, dopaminergic system appears to serve as a tonic, sustaining influence on motor activity, rather than participating on specific movements.&lt;br&gt;

2. Striatum: Normally, the striatum is connected to the substantia nigra by neurons that secrete the inhibitory transmitter GABA at their termini in the substantia nigra. In turn, cells of the substantia nigra sends neurons back to the striatum secreting the inhibitory transmitter dopamine. This mutual inhibitory pathway normally maintains a degree of inhibition of the two separate areas. Nerve fibers from the cerebral cortex and thalamus secrete acetylcholine in the neostriatum, causing excitatory effects that initiate and regulate gross intentional movements of the body. In Parkinson’s disease, destruction of cells in the substantia nigra results in the degeneration of neurons responsible for secreting dopamine in the neostriatum. Thus the normal modulating inhibitory influence of dopamine on the neostriatum is significantly diminished, resulting in the parkinsonian degeneration of the control of muscle movement.&lt;br&gt;

3. Secondary parkinsonism: Parkinsonian symptoms infrequently follow viral encephalitis or multiple small vascular lesions. Drugs such as phenothiazines and haloperidol, whose major pharmacologic action is blockade of dopamine receptors in the brain, may also produce parkinsonian symptoms. These drugs should not be used in parkinsonian patients.&lt;br&gt;

'''B. Strategy of treatment'''&lt;br&gt;

In addition to an abundance of inhibitory dopaminergic neurons, the neostriatum is also rich in excitatory cholinergic neurons that oppose the action of dopamine. Many of the symptoms reflect an imbalance between the excitatory cholinergic neurons and the greatly diminished number of inhibitory dopaminergic neurons. Therapy is aimed at restoring dopamine in the basal ganglia and antagonizing the excitatory effect of cholinergic neurons, thus reestablishing the correct dopamine/acetylcholine balance.&lt;br&gt;

'''Drugs'''&lt;br&gt;

Currently available drugs offer temporary relief from the symptoms of the disorder, but do not arrest or reverse the neuronal degeneration caused by the disease.&lt;br&gt;

Levodopa is a metabolic precursor of dopamine. It restores dopamine level in the extrapyramidal centers (substantia nigra) that atrophy in parkinsonism. In patients with early disease, the number of residual dopaminergic neurons (about 20% of normal) is adequate for conversion of levodopa to dopamine. Levodopa decreases the rigidity, tremor, and other symptoms of parkinsonism. Unfortunately, with time the number of neurons decreases and the drug effects “wear off”.&lt;br&gt;

Dopamine itself does not cross the blood-brain barrier, but its immediate precursor levodopa is readily transported into the CNS and is converted to dopamine in the brain. Large doses of levodopa are required because much of the drug is decarboxylated to dopamine in the periphery, resulting in peripheral side effects (nausea, vomiting, cardiac arrhythmias, hypotension).&lt;br&gt;

The effects of levodopa on the CNS can be greatly enhanced by coadministration carbidopa, a dopamine decarboxylase inhibitor that does not cross the blood-brain barrier. Carbidopa diminishes the metabolism of levodopa in the GI tract and peripheral tissues. The addition of carbidopa lowers the dose of levodopa needed by4-to 5- fold and, consequently, decreases the severity of the side effects of peripherally formed dopamine.&lt;br&gt;

Bromocriptine, en ergotamine (an alkaloid with vasoconstrictor action) derivative, is a dopamine receptor agonist. The drug produces little response in patients who does not react to levodopa, but it is often used with levodopa in patients responding to drug therapy. &lt;br&gt;

It was accidentally discovered that the antiviral drug, amantadine, effective in the treatment of influenza, has antiparkinsonian action. It appears to enhance the synthesis, release, or reuptake of dopamine from the surviving neurons. Amantadine is less efficacious than levodopa, but it has fewer side effects. The drug has little effect on tremor but is more effective than the anticholinergics against rigidity and bradykinesia.&lt;br&gt;

The antimuscarinic agents (cyclodol, noracin etc.) are much less efficacious than levodopa and play only adjuvant role in the antiparkinsonian therapy. Blockage of the cholinergic transmission produces similar to augmentation of dopaminergic transmission (again, because of the creation of an imbalance in the dopamine/acetylcholine ratio). All these drugs can induce mood changes and produce xerostomia (dryness of the mouth) and visual problems, as do all muscarinic blockers. They interfere with gastrointestinal peristaltics, may cause urinary retention and increase in intraocular pressure.&lt;br&gt;

==Hemodialysis==
{{medical disclaimer}}
[[Image:Guy_getting_hemo.gif|thumb|300px|A person getting hemodialysis]]
Starting '''[[w:hemodialysis]]''' is often a frightening experience.  Hemodialysis machines are complicated and dialysis sessions often are punctuated by alarms.  At the beginning of dialysis and at the end of dialysis a lot of things happen.  Not knowing what it is can be anxiety provoking.  The following '''step-by-step description of hemodialysis''' will hopefully clarify some things for people starting dialysis and allow others to gain a better understanding of what dialysis entails.

==Pre-dialysis==
#Before or around the time the patient arrives for his/her scheduled session, a dialysis machine will be prepared. There are many models of dialysis machines, but typically in modern machines there will be a computer, [[w:Cathode ray tube|CRT]], a pump, and facility for disposable tubing and filters. The filters (the actual artificial kidneys) are cylindrical, clear plastic outside with the filter material visible inside (looks like thick paper). They are perhaps 15-18 [[w:inch]]es long, and 2-3 inches thick. They have tubing connectors at both ends. The technician or nurse will set up plumbing on the machine in a moderately complex pattern that has been worked out to move blood through the filter, allow for saline drip (or not), allow for various other medications/chemicals to be administered. How the plumbing is set up may vary between models of machine and the types of filters. For some filters, it is necessary to clear sterilizing fluid from the filter before connecting the patient. This is done by altering the plumbing to push [[w:saline (medicine)|saline]] through the filter, and carefully checked with a type of [[w:litmus test]].
# The pump does not directly contact the blood or fluid in the plumbing — it works by applying pressure to the tubing, then moving that pressure point around. Think of a disk with a protrusion in it. Put this into a close fitting 270 degree enclosure. Put plastic tubing between the enclosure and the disk, entering and exiting in the 90 open degrees. Now imagine the disk turning. It will put pressure on the tubing, and the pressure point will roll around through the 270 degrees, forcing the fluid to move (see also [[w:Peristaltic pump]]). It is characteristic of dialysis machines that most of the blood out of the patients body at any given time is visible. This facilitates troubleshooting, particularly detection of clotting.
#The patient arrives and is carefully weighed. Standing and sitting blood pressures are taken. Temperature is taken.
#Access is set up. For patients with a [[w:fistula]] (a surgical modification to an arm or leg vein to make it more robust, and therefore usable for high capacity blood movement required by dialysis) this means inserting two large gauge needles into the fistula. This is painful for the patient but there are various methods of numbing the entry sites before the needles are inserted — the two most common are lignocaine (lidocaine), a local anaesthetic injected under the skin, and there is also a cream called EMLA which is applied to the skin 45 minutes before the needles are inserted. Fistulas are widely considered the desirable way to get access for hemodialysis, but they take time to set up and mature (anywhere between 5 weeks to 15 weeks). For other patients, access may be via a [[w:catheter]] installed to connect to large veins in the chest. Other arrangements can be made as well.
#When access has been set up, the patient is then connected to the preconfigured plumbing, creating a complete loop through the pump and filter.

==Dialysis==
#The pump and a timer are started. Hemodialysis is underway.
#Periodically (every half-hour, nominally) blood pressure is taken. As a practical matter, fluid is also removed during dialysis. Most dialysis patients are on moderate to severe fluid restrictive diets (in addition to other dietary restrictions), since kidney failure usually includes an inability to properly regulate fluid levels in the body. A session of hemodialysis may typically remove 2-5 [[w:kilogram]]s (5-10 pounds) of fluid from the patient. The amount of fluid to be removed is set by the dialysis nurse according to the patient's &quot;estimated dry weight.&quot; This is a weight that the care staff believes represents what the patient should weigh without fluid built up because of kidney failure. Removing this much fluid can cause or exacerbate [[w:low blood pressure]]. Monitoring is intended to detect this before it becomes too severe. Low blood pressure can cause cramping or loss of consciousness. Often this is temporary and passes after the head is placed down ([[w:Trendelenburg position]]) for a short time.

==Post-dialysis==
#At the end of the prescribed time, the patient is disconnected from the plumbing - blood lines (which is removed and discarded, except perhaps for the filter, which may be sterilized and reused for the same patient at a later date). Needle wounds (in case of fistula) are bandaged with gauze, held for 10 to 15 minutes with direct pressure to stop bleeding, and then taped in place. The process is similar to getting blood drawn, only it is lengthier, and more fluid or blood is lost.
#Temperature, standing and sitting blood pressure, and weight are all measured again. Temperature changes may indicate infection. BP discussed above. Weighing is to confirm the removal of the desired amount of fluid.
#Care staff verifies that the patient is in condition suitable for leaving. The patient must be able to stand (if previously able), maintain a reasonable blood pressure, and be coherent (if normally coherent). Different rules apply for in-patient treatment.

==Post-dialysis ''washout''==
Following hemodialysis, patients may experience a syndrome called &quot;washout&quot;. The patient feels weak, tremulous, and may suffer extreme fatigue. Patients report they &quot;are too tired, too weak to converse, hold a book or even a newspaper.&quot; It may also vary in intensity ranging from whole body aching, stiffness in joints and other flu-like symptoms including  headaches, nausea and loss of appetite. The syndrome may begin toward the end of treatment or minutes following the treatment. It may last 30 minutes or 12-14 hours in a dissipating form.  Patients though exhausted have difficulty falling asleep. Eating a light meal, rest and quiet help the patient cope with washout until it has 'worn away.'

==References==
{{Reflist}}

==External links==
* [https://web.archive.org/web/20051218030408/http://www.kidney.ca/english/publications/brochures/hemodialysis/hemodialysis.htm What is dialysis?] - Kidney Foundation of Canada

[[Category:Anatomy]]</text>
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  <page>
    <title>Physics Formulae</title>
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This aim of this article is to provide links to external articles on tabulations of fundamental physics formulae, laws, principles, definitions of important quantities, and their application to useful results which are frequently encountered in physical problems. As physics is such a large subject, each branch of physics requires its own article for tabuluations of equations.  

The scope is that of advanced school/ introductory degree level Physics, and beyond. For generality the language of calculus (including multiple integrals and partial differentials), vector algebra and calculus and matrices is neccersary. Tensors are also sometimes required for generality, but kept to a minimum. Only SI units and their corresponding dimensions are used (i.e. no natural/characteristic units or non-dimensional equations).


=='''Fundamental, General Quantities'''==


The quantities below recur throughout all fields of physics, characteristic to the foundational SI units. The articles which follow use this nomenclature unless otherwise stated.


{| class=&quot;wikitable&quot;
|-
! '''Quantity (Common Name/s)''' !! '''(Common) Symbol/s''' !! '''SI Units''' !! '''Dimension'''
|-
| Length, width, height, depth || &lt;math&gt; a,b,c,d,h,l \,\!&lt;/math&gt; 

&lt;math&gt;r,w,x,y,z \,\!&lt;/math&gt; 

|| m || [L]
|-
| (Spatial) Position Vector || &lt;math&gt; \mathbf{r}, \mathbf{R}, \mathbf{a}, \mathbf{d} \,\!&lt;/math&gt; || m || [L]
|-
| Area || &lt;math&gt; A,S \,\!&lt;/math&gt; || m&lt;sup&gt;2&lt;/sup&gt; || [L]&lt;sup&gt;2&lt;/sup&gt;
|-
| Area Vector || &lt;math&gt; \mathbf{A} = A\mathbf{\hat{n}},\mathbf{S}= S\mathbf{\hat{n}} \,\!&lt;/math&gt; || m&lt;sup&gt;2&lt;/sup&gt; || [L]&lt;sup&gt;2&lt;/sup&gt;
|-
| Volume || &lt;math&gt; V \,\!&lt;/math&gt; || m&lt;sup&gt;3&lt;/sup&gt; || [L]&lt;sup&gt;3&lt;/sup&gt;
|-
| Plane Angle || &lt;math&gt; \alpha, \beta, \gamma, \theta, \phi, \chi \,\!&lt;/math&gt; || rad || dimensionless
|-
| Solid Angle || &lt;math&gt; \omega, \Omega \,\!&lt;/math&gt; || sr|| dimensionless
|-
| Angular Position, 

Angle (of Rotation) 

|| &lt;math&gt; \theta \,\!&lt;/math&gt; || rad || dimensionless
|-

| Time || &lt;math&gt; t \,\!&lt;/math&gt; || s || [T]
|-
| Mass || &lt;math&gt; m \,\!&lt;/math&gt; || kg || [M]
|-
| Temperature || &lt;math&gt; \theta \,\!&lt;/math&gt; || K || [Te]
|-

|}

=='''Derived General Quantities'''==


Many quantities in the classical physics section, such as mass, momentum, force, energy are are also continuously used throughout all fields of physics, with appropriate modifications. They are not fundamental as they can be derived.

=='''Main Articles'''==


''' ''General Physics'' '''

[[/Conservation and Continuity Equations/]]


''' ''Classical Physics'' '''


[[/Classical Mechanics Formulae/]]

[[/Equations for Properties of Matter/]] 

[[/Thermodynamics Formulae/]]

[[/Waves Formulae/]]

[[/Gravitation Formulae/]]


''' ''Foundational Modern Physics'' '''


[[/Electromagnetism Formulae/]]

[[/Electric Circuits Formulae/]]

[[/Equations of Light/]]


''' ''Advanced Modern Physics'' '''


[[/Quantum Mechanics Formulae/]]

[[/Special Relativity Formulae/]]

[[/General Relativity Formulae/]]

[[/Astronomy and Cosmology Formulae/]]

[[/Particle Physics Formulae/]]

==References==
{{Reflist}}
* {{cite book|author1=David Halliday|author2=Robert Resnick|author3=Jearl Walker|title=Fundamentals of Physics|url=http://books.google.com/books?id=y7lYHi4OZB0C|accessdate=31 March 2011|date=15 March 2010|publisher=John Wiley and Sons|isbn=9780470469118}}

* {{cite book|author=Nouredine Zettili|title=Quantum mechanics: concepts and applications|url=http://books.google.com/books?id=6jXlpJCSz98C|accessdate=31 March 2011|year=2009|publisher=John Wiley and Sons|isbn=9780470026786}}

==External links==
*[http://www.xs4all.nl/~johanw/contents.html Physics formulae] at xs4all.nl
*[https://www.pw.live/physics-formula Physics formulas] at Physics Wallah

{{DEFAULTSORT:List Of Elementary Physics Formulae}}

[[fr:Formules de physique]]
[[fa:فهرست فرمولهای مقدماتی فیزیک]]
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    <title>The necessities in Digital Design</title>
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== ''' Number Systems '''== 

=== ''' Binary Representation '''=== 
* Binary Numbers ([[Media:DD1.1.A.BinaryNum.20130918.pdf|A.pdf]])
* Hexadecimal Numbers ([[Media:DD1.2.A.HexaNum.20130918.pdf|A.pdf]])
* Other Codes ([[Media:DD1.3A.Code.20250329.pdf|A.pdf]])

=== ''' Binary Arithmetic '''=== 
* Binary Arithmetic ([[Media:DD1.4.A.BinaryArith.20150425.pdf|A.pdf]])
* BCD Arithmetic ([[Media:DD1.5.A.BCDArith.20130918.pdf|A.pdf]])

=== ''' C Program Examples '''=== 
* Binary Numbers in C programs ([[Media:DD1.6.A.BNumInC.20140103.pdf|A.pdf]])
* Binary Addition in C programs ([[Media:DD1.7.A.BArithInC.20140103.pdf|A.pdf]])
&lt;/br&gt;

* Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|C.pdf]])
&lt;/br&gt;

=== ''' Floating Point Numbers '''=== 

* Floating Point Representations  ([[Media:CDesign.5.A.FPoint.20140121.pdf|5A.pdf]])&lt;/br&gt;
:: See [http://www.iro.umontreal.ca/~aboulham/F1214/Session%206Arithm/Floating_Point_Numbers.pdf Floating Point Overview]
:: See [http://www.cs.auckland.ac.nz/~patrice/210-2006/210%20LN04_2.pdf Offset Binary Overview]
:: See [http://www.intersil.com/content/dam/Intersil/documents/an96/an9657.pdf Offset Binary &amp; Sin / Cosine]
:: See [http://www.ee.ic.ac.uk/hp/staff/dmb/courses/dig2/4_Analog.pdf Offset Binary &amp; ADC / DAC]
&lt;/br&gt;

=== ''' Interfacing Digital and Analog Signals '''=== 
* Sampling and Quantization   ([[Media:DD1.10.A.SampleQuant.20150425.pdf|A.pdf]])
* Digital-to-Analog Conversion  ([[Media:DD1.8.A.DAC.20140208.pdf|A.pdf]])
* Analog-to-Digital Conversion ([[Media:DD1.9.A.DAC.20140208.pdf|A.pdf]])
&lt;/br&gt;

== '''Combinational Circuits'''==

=== ''' Design '''=== 
* Boolean Algebra ([[Media:DD2.A1.BAlgebra.20250503.pdf|A1.pdf]])
* Truth Tables ([[Media:DD2.A2.TTable.20250424.pdf|A2.pdf]])
* K-Map ([[Media:DD2.A3.KMap.20250424.pdf|A3.pdf]])
* Design Examples ([[Media:DD2.A4.CombEx.20250414.pdf|A4.pdf]])
&lt;/br&gt;

=== ''' Components '''=== 
* Decoder ([[Media:DD2.B.1.Decoder.20130928.pdf|B1.pdf]])
* Encoder ([[Media:DD2.B.2.Encoder.20130917.pdf|B2.pdf]])
* Multiplexer ([[Media:DD2.B.3.Multiplexer.20130928.pdf|B3.pdf]])
* Adder ([[Media:DD2.B.4..Adder.20131007.pdf|B4.pdf]], [[Media:Fa.sch.20131002.pdf|fa.sch.pdf]], [[Media:Adder4.sch.20131002.pdf|adder4.sch.pdf]])
&lt;/br&gt;

=== ''' Design Metric '''=== 
* Noise Margin ([[Media:DD2.C1.NoiseMargin.20250415.pdf|C1.pdf]])
&lt;/br&gt;

== '''Sequential Circuits'''==

=== ''' Design '''=== 
* Types of Flip-Flops ([[Media:CDesign.1.A.FF.20130412.pdf |1A.pdf]])&lt;/br&gt;
* Latches and Flipflops ([[Media:DD3.A.1.LatchFF.20160308.pdf|A1.pdf]])
* State Transition Table ([[Media:DD3.A.2.pdf|A2.pdf]])
* FSM (Finite State Machine) ([[Media:DD3.A.3.FSM.20131030.pdf|A3.pdf]])
&lt;/br&gt;

* The Classic FF Design ([[Media:DD3.A.6.ClassicFF.20131126.pdf|A7.pdf]])
* The Modern FF Design ([[Media:DD3.A.6.ClassicFF.20131204.2.pdf|A8.pdf]])
&lt;/br&gt;

=== ''' Components '''=== 
* Latches and Flip-flops ([[Media:DD3.B.1.LatchFF.20131008.pdf|B1.pdf]])
* Registers ([[Media:DD3.B.2.Register.20150326.pdf|B2.pdf]], [[Media:Register.20131118.pdf|register.pdf]])
* Counters ([[Media:DD3.B.2.Counter.20150420.pdf|B3.pdf]])
&lt;/br&gt;

=== ''' Timing Analysis '''=== 
* Metastability ([[Media:DD3.A.4.MetaState.20131030.pdf|A4.pdf]])
* Flip-flop Timing ([[Media:DD3.A5.FFTiming.20260518.pdf|A5.pdf]])
* SR Latch Forbidden State ([[Media:DD3.A.5.ForbiddenState.20131030.pdf|A6.pdf]])
&lt;/br&gt;

* FF Min Max Timing Constraints ([[Media:CArch.MinMaxTiming.20131121.pdf |pdf]])
* FF Clock Skew Timing Constraints ([[Media:CArch.ClockSkew.20131121.pdf |pdf]])

* Synchronizer ([[Media:CArch.Synchronizer.20131216.pdf |pdf]])
* Resolution Time Analysis  ([[Media:CArch.Resolution.20131216.pdf |pdf]])
&lt;/br&gt;

== '''Finite State Machine'''==

* FSM State Encoding 
* FSM Types : Mealy and Moore Machines
* FSM Example ([[Media:CArch.2.A.FSMExample.20141018.pdf |pdf]])
&lt;/br&gt;

== '''Array  Devices''' ==

=== ''' Memory Arrays '''=== 
* RAM 
** RAM Structure ([[Media:DD4.A.1.RAM.20131111.pdf|A.pdf]])
** RAM Timing ([[Media:DD4.B.1.RAMTiming.20131130.pdf|B.pdf]])
** FPGA RAM ([[Media:DD4.C.1.FPGARAM.20160513.pdf|C.pdf]])

* ROM
&lt;/br&gt;

=== ''' Logic Arrays '''=== 
* PLA 
* PAL
* PLD
* FPGA
** FPGA Structure
** FPGA Configuration ([[Media:DD4.C.1.FPGAConf.20131130.pdf|B.pdf]])
&lt;/br&gt;
&lt;/br&gt; 
[http://www.ece.cmu.edu/~ece548/localcpy/sramop.pdf  Synchronous SRAM Timing] &lt;/br&gt;
[http://www.micron.com/~/media/Documents/Products/Technical%20Note/DRAM/tn4529.pdf Asynchronous SRAM Timing]&lt;/br&gt;
[http://www.ece.cmu.edu/~ece548/localcpy/dramop.pdf DRAM Timing] &lt;/br&gt;

[http://www.ece.unm.edu/~jimp/415/slides/fpga_arch1.pdf FPGA Architectures] &lt;/br&gt;
[http://www.engr.siu.edu/~haibo/ece428/notes/ece428_fpgaarch.pdf CPLD &amp; FPGA] &lt;/br&gt;
&lt;/br&gt;

== ''' RTL Design Techniques''' ==
&lt;/br&gt;

''' Design Methodology '''
&lt;/br&gt;


''' Synthesis '''
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;

== '''Logic Families and IOs''' ==
* BJT Based 
:: DTL (Diode-Transistor Logic)
:: TTL (Transistor-Transistor Logic)
:: ECL (Emitter-Coupled Logic)
 
* MOS Based
:: CMOS (Complementary MOS)
:: Pseudo-nMOS 
:: Transmission Gate
:: BiCMOS (Bipolr + CMOS)
 
* Dynamic CMOS 
:: Domino
:: Clocked-CMOS (C&lt;sup&gt;2&lt;/sup&gt;MOS)
&lt;/br&gt; 

* Modern I/O Standards
:: TTL and LVTTL (Low Voltage TTL)
:: CMOS and LVCMOS (Low Voltage CMOS)
:: SSTL (Stub Series Terminated Logic)
:: HSTL (High Speed Tranceiver Logic)
:: LVDS (Low Voltage Differential Signaling) 

&lt;/br&gt;

* Wikipedia Pages for Logic Families ([[Media:Logic Families.wiki.20140812.pdf|A.pdf]])
&lt;/br&gt;
&lt;/br&gt;

See also &lt;/br&gt;
&lt;[[The necessities in Computer Design]]&gt;  &lt;/br&gt;
&lt;[[The necessities in Computer Architecture]]&gt;  &lt;/br&gt;
&lt;[[The necessities in Computer Organization]]&gt;  &lt;/br&gt;


&lt;/br&gt; &lt;/br&gt;

&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

== '''Old''' ==
'''Until 2011.12''' 
'''Chapter 1. Binary Numbers'''
* 1.1 Binary Numbers([[Media:BinaryNumbers.1.A.pdf|pdf]])

''' Minterm, Maxterm, HW '''
* 1.1 Lecture01([[Media:DigitalDesign.20110922.pdf|pdf]])

''' Overflow HW '''
* Overflow Table([[Media:Overflow table.20110924.pdf|pdf]])

''' K-Map '''
* K-Map([[Media:DigitalDesign.20110926.pdf|pdf]])

''' Binary Adder '''
* Binary Adder (C, S) ([[Media:DigitalDesign.20110929.pdf|pdf]])
* Overflow detection circuit (V) ([[Media:HW Overflow20111001.pdf|pdf]])

''' BCD to Ex3 Code Coversion, Dont' Care '''
* BCD to Ex3 Code Conversion ([[Media:DigitalDesign.20111006.pdf|pdf]])

''' Prime Implicant, Dont' Care '''
* Prime Implicant, Don't Care ([[Media:DigitalDesign.20111010.pdf|pdf]])
* HW 3.6 - explain the method of combining 0's and X's

''' Multiplexer / Demultiplexer '''
* Multiplexer ([[Media:DigitalDesign.20111024.pdf|pdf]])
* HW (TBD)

''' Flip Flop / Latch '''
* FF &amp; Latch ([[Media:DigitalDesign.20111027.pdf|pdf]])
* FF &amp; Latch HW ([[Media:DigitalDesign (HW).20111027.pdf|pdf]])
* Gated D Latch &amp; Master-Slave D FlipFlop ([[Media:DigitalDesign.20111031.pdf|pdf]])
* HW (Forbidden state and Indeterminate state) ([[Media:DigitalDesign (HW).20111102.pdf|pdf]]) (note in #2, S' R' instead of S R)
* Classical Edge Triggered D FlipFlop ([[Media:DigitalDesign.20111112.pdf|pdf]])
* HW (addition in SW and HW) ([[Media:DigitalDesign (HW).20111112.pdf|pdf]]) 
* FSM1  ([[Media:DigitalDesign.FSM1.20111117.pdf|pdf]]) 
* FSM2  ([[Media:DigitalDesign.FSM2.20111117.pdf|pdf]]) 
* HW (FSM Waveforms) ([[Media:DigitalDesign (HW).20111118.pdf|pdf]]) 
 
 
''' Counter '''
* Sychronous Counter ([[Media:DigitalDesign.20111121.pdf|pdf]])
* Ripple Counter, Multiplexer, Tri-state buffer([[Media:DigitalDesign.20111124.pdf|pdf]])
* Register ([[Media:DigitalDesign.register.20111201.pdf|pdf]])
* Timing ([[Media:DigitalDesign.timing.20111201.pdf|pdf]])
* HW (Multiplexer, Shift Register) ([[Media:DigitalDesign (HW).20111201.pdf|pdf]])
* Universal Shift Register, Memory Cell ([[Media:DigitalDesign.20111206.pdf|pdf]])
* HW (Bit Serial Adder) ([[Media:DigitalDesign (HW).20111206.pdf|pdf]])
 
 
''' Memory '''
* Memory ([[Media:DigitalDesign.20111208.pdf|pdf]])

''' Comparator, Multiplier '''
* Comparator, Multiplier ([[Media:DigitalDesign.20111219.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111219.draw.pdf|2.pdf]])
 
 
'''Multiplexer based design method '''
* Multiplexer Design Method ([[Media:DigitalDesign.20111221.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111221.draw.pdf|2.pdf]])


midterm result ([[Media:MidReult.20111027.pdf|pdf]])

 
* Edge Triggered Flip Flop ([[Media:EdgeTrigFF.20111224.pdf|pdf]])
* FF Timing ([[Media:FFTiming.20111203.pdf|pdf]])


&lt;/br&gt; &lt;/br&gt;
'''Until 2013.07''' 

''' Number Systems '''
* Binary Numbers ([[Media:DD.1.A.BinNum.20130309.pdf|A.pdf]])
* Hexadecimal Numbers ([[Media:DD.1.B.HexaNum.20130417.pdf|B.pdf]])
* Numbers in C programs ([[Media:DD.1.C.CNum.20130309.pdf|C.pdf]])
* Codes ([[Media:DD.1.D.Coding.20130319.pdf|pdf]])
&lt;/br&gt;

&lt;/br&gt;
* Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|pdf]])
&lt;/br&gt;

''' Combinational Circuits '''
* Truth Tables and Boolean Functions ([[Media:DD.2.A.TTable.20130325.pdf|2A.pdf]])&lt;/br&gt;
* K-Map ([[Media:DD.2.A.KMap.20130329.pdf|2B.pdf]])&lt;/br&gt;
* Binary Addition in C ([[Media:DD.2.C.BAinC.20130329.pdf|2.C.pdf]])&lt;/br&gt;
* Binary Arithmetic ([[Media:DD.2.D.BAri.2013.pdf|2.D.pdf]])&lt;/br&gt;
* Boolean Algebra ([[Media:DD.2.E.BAlgebra.20130419.pdf|2.E.pdf]])&lt;/br&gt;
&lt;/br&gt;

''' Sequential Circuits '''
* Latches and Flip-flops ([[Media:DD.3.A.LatchFF.20130413.pdf|3A.pdf]])&lt;/br&gt;
* FSM (Finite State Machine) ([[Media:DD.3.B.FSM.20130417.pdf|3B.pdf]])&lt;/br&gt;
* SR Latch Forbidden State ([[Media:DD.3.C.FState.20130413.pdf|3C.pdf]])&lt;/br&gt;
* Flip-flop Timing ([[Media:DD.3.D.Timing.20130413.pdf|3D.pdf]])&lt;/br&gt;
* Metastability ([[Media:DD.3.E.MetaState.20130628.pdf|3E.pdf]])&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;

See also &lt;/br&gt;
&quot;[[The necessities in Computer Design]]&quot;  &lt;/br&gt;
&quot;[[The necessities in Computer Architecture]]&quot;  &lt;/br&gt;

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
      <sha1>c40rz9ix91igofp47ay2x27jq3zrh8i</sha1>
    </revision>
    <revision>
      <id>2810570</id>
      <parentid>2810568</parentid>
      <timestamp>2026-05-20T08:26:54Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/*  Timing Analysis  */</comment>
      <origin>2810570</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9835" sha1="7cdn9fyotrpr9n0avx10uu4g7clia2o" xml:space="preserve">
== ''' Number Systems '''== 

=== ''' Binary Representation '''=== 
* Binary Numbers ([[Media:DD1.1.A.BinaryNum.20130918.pdf|A.pdf]])
* Hexadecimal Numbers ([[Media:DD1.2.A.HexaNum.20130918.pdf|A.pdf]])
* Other Codes ([[Media:DD1.3A.Code.20250329.pdf|A.pdf]])

=== ''' Binary Arithmetic '''=== 
* Binary Arithmetic ([[Media:DD1.4.A.BinaryArith.20150425.pdf|A.pdf]])
* BCD Arithmetic ([[Media:DD1.5.A.BCDArith.20130918.pdf|A.pdf]])

=== ''' C Program Examples '''=== 
* Binary Numbers in C programs ([[Media:DD1.6.A.BNumInC.20140103.pdf|A.pdf]])
* Binary Addition in C programs ([[Media:DD1.7.A.BArithInC.20140103.pdf|A.pdf]])
&lt;/br&gt;

* Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|C.pdf]])
&lt;/br&gt;

=== ''' Floating Point Numbers '''=== 

* Floating Point Representations  ([[Media:CDesign.5.A.FPoint.20140121.pdf|5A.pdf]])&lt;/br&gt;
:: See [http://www.iro.umontreal.ca/~aboulham/F1214/Session%206Arithm/Floating_Point_Numbers.pdf Floating Point Overview]
:: See [http://www.cs.auckland.ac.nz/~patrice/210-2006/210%20LN04_2.pdf Offset Binary Overview]
:: See [http://www.intersil.com/content/dam/Intersil/documents/an96/an9657.pdf Offset Binary &amp; Sin / Cosine]
:: See [http://www.ee.ic.ac.uk/hp/staff/dmb/courses/dig2/4_Analog.pdf Offset Binary &amp; ADC / DAC]
&lt;/br&gt;

=== ''' Interfacing Digital and Analog Signals '''=== 
* Sampling and Quantization   ([[Media:DD1.10.A.SampleQuant.20150425.pdf|A.pdf]])
* Digital-to-Analog Conversion  ([[Media:DD1.8.A.DAC.20140208.pdf|A.pdf]])
* Analog-to-Digital Conversion ([[Media:DD1.9.A.DAC.20140208.pdf|A.pdf]])
&lt;/br&gt;

== '''Combinational Circuits'''==

=== ''' Design '''=== 
* Boolean Algebra ([[Media:DD2.A1.BAlgebra.20250503.pdf|A1.pdf]])
* Truth Tables ([[Media:DD2.A2.TTable.20250424.pdf|A2.pdf]])
* K-Map ([[Media:DD2.A3.KMap.20250424.pdf|A3.pdf]])
* Design Examples ([[Media:DD2.A4.CombEx.20250414.pdf|A4.pdf]])
&lt;/br&gt;

=== ''' Components '''=== 
* Decoder ([[Media:DD2.B.1.Decoder.20130928.pdf|B1.pdf]])
* Encoder ([[Media:DD2.B.2.Encoder.20130917.pdf|B2.pdf]])
* Multiplexer ([[Media:DD2.B.3.Multiplexer.20130928.pdf|B3.pdf]])
* Adder ([[Media:DD2.B.4..Adder.20131007.pdf|B4.pdf]], [[Media:Fa.sch.20131002.pdf|fa.sch.pdf]], [[Media:Adder4.sch.20131002.pdf|adder4.sch.pdf]])
&lt;/br&gt;

=== ''' Design Metric '''=== 
* Noise Margin ([[Media:DD2.C1.NoiseMargin.20250415.pdf|C1.pdf]])
&lt;/br&gt;

== '''Sequential Circuits'''==

=== ''' Design '''=== 
* Types of Flip-Flops ([[Media:CDesign.1.A.FF.20130412.pdf |1A.pdf]])&lt;/br&gt;
* Latches and Flipflops ([[Media:DD3.A.1.LatchFF.20160308.pdf|A1.pdf]])
* State Transition Table ([[Media:DD3.A.2.pdf|A2.pdf]])
* FSM (Finite State Machine) ([[Media:DD3.A.3.FSM.20131030.pdf|A3.pdf]])
&lt;/br&gt;

* The Classic FF Design ([[Media:DD3.A.6.ClassicFF.20131126.pdf|A7.pdf]])
* The Modern FF Design ([[Media:DD3.A.6.ClassicFF.20131204.2.pdf|A8.pdf]])
&lt;/br&gt;

=== ''' Components '''=== 
* Latches and Flip-flops ([[Media:DD3.B.1.LatchFF.20131008.pdf|B1.pdf]])
* Registers ([[Media:DD3.B.2.Register.20150326.pdf|B2.pdf]], [[Media:Register.20131118.pdf|register.pdf]])
* Counters ([[Media:DD3.B.2.Counter.20150420.pdf|B3.pdf]])
&lt;/br&gt;

=== ''' Timing Analysis '''=== 
* Metastability ([[Media:DD3.A.4.MetaState.20131030.pdf|A4.pdf]])
* Flip-flop Timing ([[Media:DD3.A5.FFTiming.20260519.pdf|A5.pdf]])
* SR Latch Forbidden State ([[Media:DD3.A.5.ForbiddenState.20131030.pdf|A6.pdf]])
&lt;/br&gt;

* FF Min Max Timing Constraints ([[Media:CArch.MinMaxTiming.20131121.pdf |pdf]])
* FF Clock Skew Timing Constraints ([[Media:CArch.ClockSkew.20131121.pdf |pdf]])

* Synchronizer ([[Media:CArch.Synchronizer.20131216.pdf |pdf]])
* Resolution Time Analysis  ([[Media:CArch.Resolution.20131216.pdf |pdf]])
&lt;/br&gt;

== '''Finite State Machine'''==

* FSM State Encoding 
* FSM Types : Mealy and Moore Machines
* FSM Example ([[Media:CArch.2.A.FSMExample.20141018.pdf |pdf]])
&lt;/br&gt;

== '''Array  Devices''' ==

=== ''' Memory Arrays '''=== 
* RAM 
** RAM Structure ([[Media:DD4.A.1.RAM.20131111.pdf|A.pdf]])
** RAM Timing ([[Media:DD4.B.1.RAMTiming.20131130.pdf|B.pdf]])
** FPGA RAM ([[Media:DD4.C.1.FPGARAM.20160513.pdf|C.pdf]])

* ROM
&lt;/br&gt;

=== ''' Logic Arrays '''=== 
* PLA 
* PAL
* PLD
* FPGA
** FPGA Structure
** FPGA Configuration ([[Media:DD4.C.1.FPGAConf.20131130.pdf|B.pdf]])
&lt;/br&gt;
&lt;/br&gt; 
[http://www.ece.cmu.edu/~ece548/localcpy/sramop.pdf  Synchronous SRAM Timing] &lt;/br&gt;
[http://www.micron.com/~/media/Documents/Products/Technical%20Note/DRAM/tn4529.pdf Asynchronous SRAM Timing]&lt;/br&gt;
[http://www.ece.cmu.edu/~ece548/localcpy/dramop.pdf DRAM Timing] &lt;/br&gt;

[http://www.ece.unm.edu/~jimp/415/slides/fpga_arch1.pdf FPGA Architectures] &lt;/br&gt;
[http://www.engr.siu.edu/~haibo/ece428/notes/ece428_fpgaarch.pdf CPLD &amp; FPGA] &lt;/br&gt;
&lt;/br&gt;

== ''' RTL Design Techniques''' ==
&lt;/br&gt;

''' Design Methodology '''
&lt;/br&gt;


''' Synthesis '''
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;

== '''Logic Families and IOs''' ==
* BJT Based 
:: DTL (Diode-Transistor Logic)
:: TTL (Transistor-Transistor Logic)
:: ECL (Emitter-Coupled Logic)
 
* MOS Based
:: CMOS (Complementary MOS)
:: Pseudo-nMOS 
:: Transmission Gate
:: BiCMOS (Bipolr + CMOS)
 
* Dynamic CMOS 
:: Domino
:: Clocked-CMOS (C&lt;sup&gt;2&lt;/sup&gt;MOS)
&lt;/br&gt; 

* Modern I/O Standards
:: TTL and LVTTL (Low Voltage TTL)
:: CMOS and LVCMOS (Low Voltage CMOS)
:: SSTL (Stub Series Terminated Logic)
:: HSTL (High Speed Tranceiver Logic)
:: LVDS (Low Voltage Differential Signaling) 

&lt;/br&gt;

* Wikipedia Pages for Logic Families ([[Media:Logic Families.wiki.20140812.pdf|A.pdf]])
&lt;/br&gt;
&lt;/br&gt;

See also &lt;/br&gt;
&lt;[[The necessities in Computer Design]]&gt;  &lt;/br&gt;
&lt;[[The necessities in Computer Architecture]]&gt;  &lt;/br&gt;
&lt;[[The necessities in Computer Organization]]&gt;  &lt;/br&gt;


&lt;/br&gt; &lt;/br&gt;

&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

== '''Old''' ==
'''Until 2011.12''' 
'''Chapter 1. Binary Numbers'''
* 1.1 Binary Numbers([[Media:BinaryNumbers.1.A.pdf|pdf]])

''' Minterm, Maxterm, HW '''
* 1.1 Lecture01([[Media:DigitalDesign.20110922.pdf|pdf]])

''' Overflow HW '''
* Overflow Table([[Media:Overflow table.20110924.pdf|pdf]])

''' K-Map '''
* K-Map([[Media:DigitalDesign.20110926.pdf|pdf]])

''' Binary Adder '''
* Binary Adder (C, S) ([[Media:DigitalDesign.20110929.pdf|pdf]])
* Overflow detection circuit (V) ([[Media:HW Overflow20111001.pdf|pdf]])

''' BCD to Ex3 Code Coversion, Dont' Care '''
* BCD to Ex3 Code Conversion ([[Media:DigitalDesign.20111006.pdf|pdf]])

''' Prime Implicant, Dont' Care '''
* Prime Implicant, Don't Care ([[Media:DigitalDesign.20111010.pdf|pdf]])
* HW 3.6 - explain the method of combining 0's and X's

''' Multiplexer / Demultiplexer '''
* Multiplexer ([[Media:DigitalDesign.20111024.pdf|pdf]])
* HW (TBD)

''' Flip Flop / Latch '''
* FF &amp; Latch ([[Media:DigitalDesign.20111027.pdf|pdf]])
* FF &amp; Latch HW ([[Media:DigitalDesign (HW).20111027.pdf|pdf]])
* Gated D Latch &amp; Master-Slave D FlipFlop ([[Media:DigitalDesign.20111031.pdf|pdf]])
* HW (Forbidden state and Indeterminate state) ([[Media:DigitalDesign (HW).20111102.pdf|pdf]]) (note in #2, S' R' instead of S R)
* Classical Edge Triggered D FlipFlop ([[Media:DigitalDesign.20111112.pdf|pdf]])
* HW (addition in SW and HW) ([[Media:DigitalDesign (HW).20111112.pdf|pdf]]) 
* FSM1  ([[Media:DigitalDesign.FSM1.20111117.pdf|pdf]]) 
* FSM2  ([[Media:DigitalDesign.FSM2.20111117.pdf|pdf]]) 
* HW (FSM Waveforms) ([[Media:DigitalDesign (HW).20111118.pdf|pdf]]) 
 
 
''' Counter '''
* Sychronous Counter ([[Media:DigitalDesign.20111121.pdf|pdf]])
* Ripple Counter, Multiplexer, Tri-state buffer([[Media:DigitalDesign.20111124.pdf|pdf]])
* Register ([[Media:DigitalDesign.register.20111201.pdf|pdf]])
* Timing ([[Media:DigitalDesign.timing.20111201.pdf|pdf]])
* HW (Multiplexer, Shift Register) ([[Media:DigitalDesign (HW).20111201.pdf|pdf]])
* Universal Shift Register, Memory Cell ([[Media:DigitalDesign.20111206.pdf|pdf]])
* HW (Bit Serial Adder) ([[Media:DigitalDesign (HW).20111206.pdf|pdf]])
 
 
''' Memory '''
* Memory ([[Media:DigitalDesign.20111208.pdf|pdf]])

''' Comparator, Multiplier '''
* Comparator, Multiplier ([[Media:DigitalDesign.20111219.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111219.draw.pdf|2.pdf]])
 
 
'''Multiplexer based design method '''
* Multiplexer Design Method ([[Media:DigitalDesign.20111221.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111221.draw.pdf|2.pdf]])


midterm result ([[Media:MidReult.20111027.pdf|pdf]])

 
* Edge Triggered Flip Flop ([[Media:EdgeTrigFF.20111224.pdf|pdf]])
* FF Timing ([[Media:FFTiming.20111203.pdf|pdf]])


&lt;/br&gt; &lt;/br&gt;
'''Until 2013.07''' 

''' Number Systems '''
* Binary Numbers ([[Media:DD.1.A.BinNum.20130309.pdf|A.pdf]])
* Hexadecimal Numbers ([[Media:DD.1.B.HexaNum.20130417.pdf|B.pdf]])
* Numbers in C programs ([[Media:DD.1.C.CNum.20130309.pdf|C.pdf]])
* Codes ([[Media:DD.1.D.Coding.20130319.pdf|pdf]])
&lt;/br&gt;

&lt;/br&gt;
* Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|pdf]])
&lt;/br&gt;

''' Combinational Circuits '''
* Truth Tables and Boolean Functions ([[Media:DD.2.A.TTable.20130325.pdf|2A.pdf]])&lt;/br&gt;
* K-Map ([[Media:DD.2.A.KMap.20130329.pdf|2B.pdf]])&lt;/br&gt;
* Binary Addition in C ([[Media:DD.2.C.BAinC.20130329.pdf|2.C.pdf]])&lt;/br&gt;
* Binary Arithmetic ([[Media:DD.2.D.BAri.2013.pdf|2.D.pdf]])&lt;/br&gt;
* Boolean Algebra ([[Media:DD.2.E.BAlgebra.20130419.pdf|2.E.pdf]])&lt;/br&gt;
&lt;/br&gt;

''' Sequential Circuits '''
* Latches and Flip-flops ([[Media:DD.3.A.LatchFF.20130413.pdf|3A.pdf]])&lt;/br&gt;
* FSM (Finite State Machine) ([[Media:DD.3.B.FSM.20130417.pdf|3B.pdf]])&lt;/br&gt;
* SR Latch Forbidden State ([[Media:DD.3.C.FState.20130413.pdf|3C.pdf]])&lt;/br&gt;
* Flip-flop Timing ([[Media:DD.3.D.Timing.20130413.pdf|3D.pdf]])&lt;/br&gt;
* Metastability ([[Media:DD.3.E.MetaState.20130628.pdf|3E.pdf]])&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;

See also &lt;/br&gt;
&quot;[[The necessities in Computer Design]]&quot;  &lt;/br&gt;
&quot;[[The necessities in Computer Architecture]]&quot;  &lt;/br&gt;

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
      <sha1>7cdn9fyotrpr9n0avx10uu4g7clia2o</sha1>
    </revision>
    <revision>
      <id>2810572</id>
      <parentid>2810570</parentid>
      <timestamp>2026-05-20T08:27:48Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/*  Timing Analysis  */</comment>
      <origin>2810572</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9835" sha1="qhma034lszyovpayoqzs9u3ay7pvwyb" xml:space="preserve">
== ''' Number Systems '''== 

=== ''' Binary Representation '''=== 
* Binary Numbers ([[Media:DD1.1.A.BinaryNum.20130918.pdf|A.pdf]])
* Hexadecimal Numbers ([[Media:DD1.2.A.HexaNum.20130918.pdf|A.pdf]])
* Other Codes ([[Media:DD1.3A.Code.20250329.pdf|A.pdf]])

=== ''' Binary Arithmetic '''=== 
* Binary Arithmetic ([[Media:DD1.4.A.BinaryArith.20150425.pdf|A.pdf]])
* BCD Arithmetic ([[Media:DD1.5.A.BCDArith.20130918.pdf|A.pdf]])

=== ''' C Program Examples '''=== 
* Binary Numbers in C programs ([[Media:DD1.6.A.BNumInC.20140103.pdf|A.pdf]])
* Binary Addition in C programs ([[Media:DD1.7.A.BArithInC.20140103.pdf|A.pdf]])
&lt;/br&gt;

* Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|C.pdf]])
&lt;/br&gt;

=== ''' Floating Point Numbers '''=== 

* Floating Point Representations  ([[Media:CDesign.5.A.FPoint.20140121.pdf|5A.pdf]])&lt;/br&gt;
:: See [http://www.iro.umontreal.ca/~aboulham/F1214/Session%206Arithm/Floating_Point_Numbers.pdf Floating Point Overview]
:: See [http://www.cs.auckland.ac.nz/~patrice/210-2006/210%20LN04_2.pdf Offset Binary Overview]
:: See [http://www.intersil.com/content/dam/Intersil/documents/an96/an9657.pdf Offset Binary &amp; Sin / Cosine]
:: See [http://www.ee.ic.ac.uk/hp/staff/dmb/courses/dig2/4_Analog.pdf Offset Binary &amp; ADC / DAC]
&lt;/br&gt;

=== ''' Interfacing Digital and Analog Signals '''=== 
* Sampling and Quantization   ([[Media:DD1.10.A.SampleQuant.20150425.pdf|A.pdf]])
* Digital-to-Analog Conversion  ([[Media:DD1.8.A.DAC.20140208.pdf|A.pdf]])
* Analog-to-Digital Conversion ([[Media:DD1.9.A.DAC.20140208.pdf|A.pdf]])
&lt;/br&gt;

== '''Combinational Circuits'''==

=== ''' Design '''=== 
* Boolean Algebra ([[Media:DD2.A1.BAlgebra.20250503.pdf|A1.pdf]])
* Truth Tables ([[Media:DD2.A2.TTable.20250424.pdf|A2.pdf]])
* K-Map ([[Media:DD2.A3.KMap.20250424.pdf|A3.pdf]])
* Design Examples ([[Media:DD2.A4.CombEx.20250414.pdf|A4.pdf]])
&lt;/br&gt;

=== ''' Components '''=== 
* Decoder ([[Media:DD2.B.1.Decoder.20130928.pdf|B1.pdf]])
* Encoder ([[Media:DD2.B.2.Encoder.20130917.pdf|B2.pdf]])
* Multiplexer ([[Media:DD2.B.3.Multiplexer.20130928.pdf|B3.pdf]])
* Adder ([[Media:DD2.B.4..Adder.20131007.pdf|B4.pdf]], [[Media:Fa.sch.20131002.pdf|fa.sch.pdf]], [[Media:Adder4.sch.20131002.pdf|adder4.sch.pdf]])
&lt;/br&gt;

=== ''' Design Metric '''=== 
* Noise Margin ([[Media:DD2.C1.NoiseMargin.20250415.pdf|C1.pdf]])
&lt;/br&gt;

== '''Sequential Circuits'''==

=== ''' Design '''=== 
* Types of Flip-Flops ([[Media:CDesign.1.A.FF.20130412.pdf |1A.pdf]])&lt;/br&gt;
* Latches and Flipflops ([[Media:DD3.A.1.LatchFF.20160308.pdf|A1.pdf]])
* State Transition Table ([[Media:DD3.A.2.pdf|A2.pdf]])
* FSM (Finite State Machine) ([[Media:DD3.A.3.FSM.20131030.pdf|A3.pdf]])
&lt;/br&gt;

* The Classic FF Design ([[Media:DD3.A.6.ClassicFF.20131126.pdf|A7.pdf]])
* The Modern FF Design ([[Media:DD3.A.6.ClassicFF.20131204.2.pdf|A8.pdf]])
&lt;/br&gt;

=== ''' Components '''=== 
* Latches and Flip-flops ([[Media:DD3.B.1.LatchFF.20131008.pdf|B1.pdf]])
* Registers ([[Media:DD3.B.2.Register.20150326.pdf|B2.pdf]], [[Media:Register.20131118.pdf|register.pdf]])
* Counters ([[Media:DD3.B.2.Counter.20150420.pdf|B3.pdf]])
&lt;/br&gt;

=== ''' Timing Analysis '''=== 
* Metastability ([[Media:DD3.A.4.MetaState.20131030.pdf|A4.pdf]])
* Flip-flop Timing ([[Media:DD3.A5.FFTiming.20260520.pdf|A5.pdf]])
* SR Latch Forbidden State ([[Media:DD3.A.5.ForbiddenState.20131030.pdf|A6.pdf]])
&lt;/br&gt;

* FF Min Max Timing Constraints ([[Media:CArch.MinMaxTiming.20131121.pdf |pdf]])
* FF Clock Skew Timing Constraints ([[Media:CArch.ClockSkew.20131121.pdf |pdf]])

* Synchronizer ([[Media:CArch.Synchronizer.20131216.pdf |pdf]])
* Resolution Time Analysis  ([[Media:CArch.Resolution.20131216.pdf |pdf]])
&lt;/br&gt;

== '''Finite State Machine'''==

* FSM State Encoding 
* FSM Types : Mealy and Moore Machines
* FSM Example ([[Media:CArch.2.A.FSMExample.20141018.pdf |pdf]])
&lt;/br&gt;

== '''Array  Devices''' ==

=== ''' Memory Arrays '''=== 
* RAM 
** RAM Structure ([[Media:DD4.A.1.RAM.20131111.pdf|A.pdf]])
** RAM Timing ([[Media:DD4.B.1.RAMTiming.20131130.pdf|B.pdf]])
** FPGA RAM ([[Media:DD4.C.1.FPGARAM.20160513.pdf|C.pdf]])

* ROM
&lt;/br&gt;

=== ''' Logic Arrays '''=== 
* PLA 
* PAL
* PLD
* FPGA
** FPGA Structure
** FPGA Configuration ([[Media:DD4.C.1.FPGAConf.20131130.pdf|B.pdf]])
&lt;/br&gt;
&lt;/br&gt; 
[http://www.ece.cmu.edu/~ece548/localcpy/sramop.pdf  Synchronous SRAM Timing] &lt;/br&gt;
[http://www.micron.com/~/media/Documents/Products/Technical%20Note/DRAM/tn4529.pdf Asynchronous SRAM Timing]&lt;/br&gt;
[http://www.ece.cmu.edu/~ece548/localcpy/dramop.pdf DRAM Timing] &lt;/br&gt;

[http://www.ece.unm.edu/~jimp/415/slides/fpga_arch1.pdf FPGA Architectures] &lt;/br&gt;
[http://www.engr.siu.edu/~haibo/ece428/notes/ece428_fpgaarch.pdf CPLD &amp; FPGA] &lt;/br&gt;
&lt;/br&gt;

== ''' RTL Design Techniques''' ==
&lt;/br&gt;

''' Design Methodology '''
&lt;/br&gt;


''' Synthesis '''
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;

== '''Logic Families and IOs''' ==
* BJT Based 
:: DTL (Diode-Transistor Logic)
:: TTL (Transistor-Transistor Logic)
:: ECL (Emitter-Coupled Logic)
 
* MOS Based
:: CMOS (Complementary MOS)
:: Pseudo-nMOS 
:: Transmission Gate
:: BiCMOS (Bipolr + CMOS)
 
* Dynamic CMOS 
:: Domino
:: Clocked-CMOS (C&lt;sup&gt;2&lt;/sup&gt;MOS)
&lt;/br&gt; 

* Modern I/O Standards
:: TTL and LVTTL (Low Voltage TTL)
:: CMOS and LVCMOS (Low Voltage CMOS)
:: SSTL (Stub Series Terminated Logic)
:: HSTL (High Speed Tranceiver Logic)
:: LVDS (Low Voltage Differential Signaling) 

&lt;/br&gt;

* Wikipedia Pages for Logic Families ([[Media:Logic Families.wiki.20140812.pdf|A.pdf]])
&lt;/br&gt;
&lt;/br&gt;

See also &lt;/br&gt;
&lt;[[The necessities in Computer Design]]&gt;  &lt;/br&gt;
&lt;[[The necessities in Computer Architecture]]&gt;  &lt;/br&gt;
&lt;[[The necessities in Computer Organization]]&gt;  &lt;/br&gt;


&lt;/br&gt; &lt;/br&gt;

&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

== '''Old''' ==
'''Until 2011.12''' 
'''Chapter 1. Binary Numbers'''
* 1.1 Binary Numbers([[Media:BinaryNumbers.1.A.pdf|pdf]])

''' Minterm, Maxterm, HW '''
* 1.1 Lecture01([[Media:DigitalDesign.20110922.pdf|pdf]])

''' Overflow HW '''
* Overflow Table([[Media:Overflow table.20110924.pdf|pdf]])

''' K-Map '''
* K-Map([[Media:DigitalDesign.20110926.pdf|pdf]])

''' Binary Adder '''
* Binary Adder (C, S) ([[Media:DigitalDesign.20110929.pdf|pdf]])
* Overflow detection circuit (V) ([[Media:HW Overflow20111001.pdf|pdf]])

''' BCD to Ex3 Code Coversion, Dont' Care '''
* BCD to Ex3 Code Conversion ([[Media:DigitalDesign.20111006.pdf|pdf]])

''' Prime Implicant, Dont' Care '''
* Prime Implicant, Don't Care ([[Media:DigitalDesign.20111010.pdf|pdf]])
* HW 3.6 - explain the method of combining 0's and X's

''' Multiplexer / Demultiplexer '''
* Multiplexer ([[Media:DigitalDesign.20111024.pdf|pdf]])
* HW (TBD)

''' Flip Flop / Latch '''
* FF &amp; Latch ([[Media:DigitalDesign.20111027.pdf|pdf]])
* FF &amp; Latch HW ([[Media:DigitalDesign (HW).20111027.pdf|pdf]])
* Gated D Latch &amp; Master-Slave D FlipFlop ([[Media:DigitalDesign.20111031.pdf|pdf]])
* HW (Forbidden state and Indeterminate state) ([[Media:DigitalDesign (HW).20111102.pdf|pdf]]) (note in #2, S' R' instead of S R)
* Classical Edge Triggered D FlipFlop ([[Media:DigitalDesign.20111112.pdf|pdf]])
* HW (addition in SW and HW) ([[Media:DigitalDesign (HW).20111112.pdf|pdf]]) 
* FSM1  ([[Media:DigitalDesign.FSM1.20111117.pdf|pdf]]) 
* FSM2  ([[Media:DigitalDesign.FSM2.20111117.pdf|pdf]]) 
* HW (FSM Waveforms) ([[Media:DigitalDesign (HW).20111118.pdf|pdf]]) 
 
 
''' Counter '''
* Sychronous Counter ([[Media:DigitalDesign.20111121.pdf|pdf]])
* Ripple Counter, Multiplexer, Tri-state buffer([[Media:DigitalDesign.20111124.pdf|pdf]])
* Register ([[Media:DigitalDesign.register.20111201.pdf|pdf]])
* Timing ([[Media:DigitalDesign.timing.20111201.pdf|pdf]])
* HW (Multiplexer, Shift Register) ([[Media:DigitalDesign (HW).20111201.pdf|pdf]])
* Universal Shift Register, Memory Cell ([[Media:DigitalDesign.20111206.pdf|pdf]])
* HW (Bit Serial Adder) ([[Media:DigitalDesign (HW).20111206.pdf|pdf]])
 
 
''' Memory '''
* Memory ([[Media:DigitalDesign.20111208.pdf|pdf]])

''' Comparator, Multiplier '''
* Comparator, Multiplier ([[Media:DigitalDesign.20111219.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111219.draw.pdf|2.pdf]])
 
 
'''Multiplexer based design method '''
* Multiplexer Design Method ([[Media:DigitalDesign.20111221.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111221.draw.pdf|2.pdf]])


midterm result ([[Media:MidReult.20111027.pdf|pdf]])

 
* Edge Triggered Flip Flop ([[Media:EdgeTrigFF.20111224.pdf|pdf]])
* FF Timing ([[Media:FFTiming.20111203.pdf|pdf]])


&lt;/br&gt; &lt;/br&gt;
'''Until 2013.07''' 

''' Number Systems '''
* Binary Numbers ([[Media:DD.1.A.BinNum.20130309.pdf|A.pdf]])
* Hexadecimal Numbers ([[Media:DD.1.B.HexaNum.20130417.pdf|B.pdf]])
* Numbers in C programs ([[Media:DD.1.C.CNum.20130309.pdf|C.pdf]])
* Codes ([[Media:DD.1.D.Coding.20130319.pdf|pdf]])
&lt;/br&gt;

&lt;/br&gt;
* Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|pdf]])
&lt;/br&gt;

''' Combinational Circuits '''
* Truth Tables and Boolean Functions ([[Media:DD.2.A.TTable.20130325.pdf|2A.pdf]])&lt;/br&gt;
* K-Map ([[Media:DD.2.A.KMap.20130329.pdf|2B.pdf]])&lt;/br&gt;
* Binary Addition in C ([[Media:DD.2.C.BAinC.20130329.pdf|2.C.pdf]])&lt;/br&gt;
* Binary Arithmetic ([[Media:DD.2.D.BAri.2013.pdf|2.D.pdf]])&lt;/br&gt;
* Boolean Algebra ([[Media:DD.2.E.BAlgebra.20130419.pdf|2.E.pdf]])&lt;/br&gt;
&lt;/br&gt;

''' Sequential Circuits '''
* Latches and Flip-flops ([[Media:DD.3.A.LatchFF.20130413.pdf|3A.pdf]])&lt;/br&gt;
* FSM (Finite State Machine) ([[Media:DD.3.B.FSM.20130417.pdf|3B.pdf]])&lt;/br&gt;
* SR Latch Forbidden State ([[Media:DD.3.C.FState.20130413.pdf|3C.pdf]])&lt;/br&gt;
* Flip-flop Timing ([[Media:DD.3.D.Timing.20130413.pdf|3D.pdf]])&lt;/br&gt;
* Metastability ([[Media:DD.3.E.MetaState.20130628.pdf|3E.pdf]])&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;

See also &lt;/br&gt;
&quot;[[The necessities in Computer Design]]&quot;  &lt;/br&gt;
&quot;[[The necessities in Computer Architecture]]&quot;  &lt;/br&gt;

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
      <sha1>qhma034lszyovpayoqzs9u3ay7pvwyb</sha1>
    </revision>
  </page>
  <page>
    <title>VHDL programming in plain view</title>
    <ns>0</ns>
    <id>121359</id>
    <revision>
      <id>2810468</id>
      <parentid>2810154</parentid>
      <timestamp>2026-05-19T17:26:33Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Data */</comment>
      <origin>2810468</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2278" sha1="tg25fctyc9t29gjnck5vlqg14ycflfb" xml:space="preserve">&lt;!----------------------------------------------------------------------&gt;

== 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]])
&lt;br&gt; 

== Versions of VHDL ==
* VHDL Versions ([[Media:VHDL.1.A.Versions.20120619.pdf|pdf]])
* VHDL Libraries ([[Media:VHDL.1.A.Libraries.20140219.pdf|pdf]])
&lt;br&gt;

== Basic Features of VHDL ==
==== Data ====
* Data Objects ([[Media:Data.Object.1A.20260519.pdf|A]], [[Media:Data.Object.1B.20260518.pdf|B]])
* Data Types ([[Media:Data.Type.2A.20260518.pdf|A]], [[Media:Data.Type.2B.20260518.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]])
&lt;br&gt;

==== Signals &amp; Variables ====
* Signals &amp; Variables ([[Media:Signal.1A.SigVar.20250614.pdf|pdf]])
* Sequential Signal Assignments ([[Media:Signal.4A.Sequential.20250612.pdf|pdf]])
* Concurrent &amp; Sequential Signal Assignments ([[Media:Signal.1.A.ConSeq.20120611.pdf|pdf]])
* Inertial &amp; Transport Delay Models ([[Media:Signal.2.A.InertTrans.20120704.pdf|pdf]])
* Simulation &amp; Synthesis ([[Media:Signal.3.A.SimSyn.20120504.pdf|pdf]])


&lt;br&gt;

==== 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]])
&lt;/br&gt;

==== Entity and Architecture ====

&lt;br&gt; 
==== Block Statement ====

&lt;br&gt; 
==== Process Statement ====

&lt;br&gt;
==== Operators ====

&lt;br&gt;
==== Assignment Statement ====

&lt;br&gt; 
==== Concurrent Statement ====

&lt;br&gt; 
==== Sequential Control Statement ====

&lt;br&gt;
==== 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]])

&lt;br&gt;
==== Procedure ====


&lt;br&gt;
==== Package ====
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:VHDL]]
[[Category:FPGA]]</text>
      <sha1>tg25fctyc9t29gjnck5vlqg14ycflfb</sha1>
    </revision>
    <revision>
      <id>2810473</id>
      <parentid>2810468</parentid>
      <timestamp>2026-05-19T17:41:26Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Data */</comment>
      <origin>2810473</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2278" sha1="58p1d20nw1qbry3az0hlv2lm8t7y69l" xml:space="preserve">&lt;!----------------------------------------------------------------------&gt;

== 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]])
&lt;br&gt; 

== Versions of VHDL ==
* VHDL Versions ([[Media:VHDL.1.A.Versions.20120619.pdf|pdf]])
* VHDL Libraries ([[Media:VHDL.1.A.Libraries.20140219.pdf|pdf]])
&lt;br&gt;

== Basic Features of VHDL ==
==== Data ====
* Data Objects ([[Media:Data.Object.1A.20260519.pdf|A]], [[Media:Data.Object.1B.20260519.pdf|B]])
* Data Types ([[Media:Data.Type.2A.20260518.pdf|A]], [[Media:Data.Type.2B.20260518.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]])
&lt;br&gt;

==== Signals &amp; Variables ====
* Signals &amp; Variables ([[Media:Signal.1A.SigVar.20250614.pdf|pdf]])
* Sequential Signal Assignments ([[Media:Signal.4A.Sequential.20250612.pdf|pdf]])
* Concurrent &amp; Sequential Signal Assignments ([[Media:Signal.1.A.ConSeq.20120611.pdf|pdf]])
* Inertial &amp; Transport Delay Models ([[Media:Signal.2.A.InertTrans.20120704.pdf|pdf]])
* Simulation &amp; Synthesis ([[Media:Signal.3.A.SimSyn.20120504.pdf|pdf]])


&lt;br&gt;

==== 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]])
&lt;/br&gt;

==== Entity and Architecture ====

&lt;br&gt; 
==== Block Statement ====

&lt;br&gt; 
==== Process Statement ====

&lt;br&gt;
==== Operators ====

&lt;br&gt;
==== Assignment Statement ====

&lt;br&gt; 
==== Concurrent Statement ====

&lt;br&gt; 
==== Sequential Control Statement ====

&lt;br&gt;
==== 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]])

&lt;br&gt;
==== Procedure ====


&lt;br&gt;
==== Package ====
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:VHDL]]
[[Category:FPGA]]</text>
      <sha1>58p1d20nw1qbry3az0hlv2lm8t7y69l</sha1>
    </revision>
    <revision>
      <id>2810545</id>
      <parentid>2810473</parentid>
      <timestamp>2026-05-20T04:10:02Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Data */</comment>
      <origin>2810545</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2278" sha1="au49hg41mvlqtatppytlabc30x2gc6r" xml:space="preserve">&lt;!----------------------------------------------------------------------&gt;

== 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]])
&lt;br&gt; 

== Versions of VHDL ==
* VHDL Versions ([[Media:VHDL.1.A.Versions.20120619.pdf|pdf]])
* VHDL Libraries ([[Media:VHDL.1.A.Libraries.20140219.pdf|pdf]])
&lt;br&gt;

== Basic Features of VHDL ==
==== Data ====
* Data Objects ([[Media:Data.Object.1A.20260519.pdf|A]], [[Media:Data.Object.1B.20260519.pdf|B]])
* Data Types ([[Media:Data.Type.2A.20260519.pdf|A]], [[Media:Data.Type.2B.20260519.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]])
&lt;br&gt;

==== Signals &amp; Variables ====
* Signals &amp; Variables ([[Media:Signal.1A.SigVar.20250614.pdf|pdf]])
* Sequential Signal Assignments ([[Media:Signal.4A.Sequential.20250612.pdf|pdf]])
* Concurrent &amp; Sequential Signal Assignments ([[Media:Signal.1.A.ConSeq.20120611.pdf|pdf]])
* Inertial &amp; Transport Delay Models ([[Media:Signal.2.A.InertTrans.20120704.pdf|pdf]])
* Simulation &amp; Synthesis ([[Media:Signal.3.A.SimSyn.20120504.pdf|pdf]])


&lt;br&gt;

==== 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]])
&lt;/br&gt;

==== Entity and Architecture ====

&lt;br&gt; 
==== Block Statement ====

&lt;br&gt; 
==== Process Statement ====

&lt;br&gt;
==== Operators ====

&lt;br&gt;
==== Assignment Statement ====

&lt;br&gt; 
==== Concurrent Statement ====

&lt;br&gt; 
==== Sequential Control Statement ====

&lt;br&gt;
==== 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]])

&lt;br&gt;
==== Procedure ====


&lt;br&gt;
==== Package ====
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:VHDL]]
[[Category:FPGA]]</text>
      <sha1>au49hg41mvlqtatppytlabc30x2gc6r</sha1>
    </revision>
  </page>
  <page>
    <title>Linux System programming in plain view</title>
    <ns>0</ns>
    <id>136794</id>
    <revision>
      <id>2810554</id>
      <parentid>2809801</parentid>
      <timestamp>2026-05-20T06:25:14Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* File System */</comment>
      <origin>2810554</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3197" sha1="ocjpfu0jnwpwnifacxordxxkupp11bx" xml:space="preserve">This course belongs to the [[Electrical &amp; Computer Engineering Studies]]

== Introduction ==
* Introduction ([[Media:SysP.Intro.20161128.pdf|pdf]])

== File System ==
* File System ([[Media:SysP.FileSystem.20251023.pdf|pdf]])
* File Pointer ([[Media:SysP..FilePointer.20161103.pdf|pdf]])
* System Calls ([[Media:SysP.File.SysCall.20161128.pdf|pdf]])
* File IO ([[Media:SysP.FileIO.20251023.pdf|pdf]])


* Copilot: File System ([[Media:glibcFileSystem.20251029-2.pdf|pdf]])
* Copilot: File Buffer ([[Media:glibcFileBuffer.20251025-2.pdf|pdf]])
* Copilot: File IO ([[Media:glibcFileIO.20251025-2.pdf|pdf]])
* Copilot: File Permission ([[Media:glibcFilePerm.20260121.pdf|pdf]])
* Copilot: File Control ([[Media:CP.FileCntl.20260428.pdf|pdf]], [[Media:CP.FileCntl.A.20260518.pdf|A]], [[Media:CP.FileCntl.B.20260504.pdf|B]], [[Media:CP.FileCntl.C.20260501.pdf|C]])

&lt;br&gt;
&lt;br&gt;

== Process ==
* Process  ([[Media:SysP.Process.20251120.pdf|pdf]])
* Fork  ([[Media:SysP.Fork.20251126.pdf|pdf]])


* Copilot: Process Information ([[Media:glibc.Process.1Info.20251101.pdf|pdf]])
* Copilot: Process Control ([[Media:glibc.Process.2Control.20251103.pdf|pdf]])
* Copilot: Process Execution ([[Media:glibc.Proc.3Exec.20251105.pdf|pdf]])
* Copilot: Process Fork ([[Media:glibc.Proc.4Fork.20251106.pdf|pdf]])
* Copilot: Process Context Switching ([[Media:glibc.Proc.5Context.20251107.pdf|pdf]])
* Copilot: Process Exec family of functions ([[Media:glibc.Proc.6ExecCall.20251112.pdf|pdf]])
* Copilot: Process Wait family of functions ([[Media:glibc.Proc.7WaitCall.20251112.pdf|pdf]])
* Copilot: Process Exit ([[Media:glibc.Proc.8Exit.20251113.pdf|pdf]])

&lt;/br&gt;

== Inter Process Communication==

=== Signal ===
* Signal  ([[Media:SysP.7.A.Signal.20121206.pdf|pdf]])


* Copilot: Signal 1. Alarm ([[Media:glibc.Signal.Alarm.20251201.pdf|pdf]])
* Copilot: Signal 2. Other Functions ([[Media:glibc.Signal.2Other.20251205.pdf|pdf]])

&lt;/br&gt;

=== Pipe ===

* Pipe ([[Media:SysP.3.A.IPC.20121115.pdf|pdf]])


* Copilot: Pipe 1. A Special File ([[Media:glibc.Pipe.File.20260307.pdf|pdf]])

&lt;/br&gt;

=== System V IPC ===

* Message Queue  ([[Media:SysP.5.A.MessageQ.20121213.pdf|pdf]])

* Shared Memory  ([[Media:SysP.8.A.SharedMem.20121227.pdf|pdf]])

* Semaphore  ([[Media:SysP.6.A.Semaphore.20251215.pdf|pdf]])

&lt;/br&gt;

* Copilot: Message Queue ([[Media:glibc.MessageQ.20251202.pdf|pdf]])
* Copilot: Shared Memory ([[Media:glibc.SharedMem.20251203.pdf|pdf]])
* Copilot: Semaphore ([[Media:glibc.Semaphore.20251215.pdf|pdf]])

&lt;/br&gt;

=== Socket ===

* Socket  ([[Media:SysP.4.A.Socket.20121122.pdf|pdf]])

&lt;/br&gt;

== Thread ==
* POSIX thread (pthread)  ([[Media:SysP.9.A.Pthread.20130225.pdf|pdf]])

==External links==
* [http://www.tldp.org/LDP/tlk/tlk.html The Linux Kernel]
* [http://www.tldp.org/LDP/lpg/lpg.html The Linux Programmer's Guide]
* [http://www.cs.cf.ac.uk/Dave/C/ Programming in C - UNIX System Calls and Subroutines using C.]
* [http://www.cs.cmu.edu/afs/cs/academic/class/15492-f07/www/pthreads.html POSIX thread (pthread) libraries]
* [https://computing.llnl.gov/tutorials/pthreads/#Thread POSIX Threads Programming]

[[Category:Linux]]
[[Category:Computer programming]]
[[Category:C programming language]]</text>
      <sha1>ocjpfu0jnwpwnifacxordxxkupp11bx</sha1>
    </revision>
    <revision>
      <id>2810556</id>
      <parentid>2810554</parentid>
      <timestamp>2026-05-20T06:26:43Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* File System */</comment>
      <origin>2810556</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3197" sha1="7s6kzcmg1egiubwxcjdpnjs813sz8kz" xml:space="preserve">This course belongs to the [[Electrical &amp; Computer Engineering Studies]]

== Introduction ==
* Introduction ([[Media:SysP.Intro.20161128.pdf|pdf]])

== File System ==
* File System ([[Media:SysP.FileSystem.20251023.pdf|pdf]])
* File Pointer ([[Media:SysP..FilePointer.20161103.pdf|pdf]])
* System Calls ([[Media:SysP.File.SysCall.20161128.pdf|pdf]])
* File IO ([[Media:SysP.FileIO.20251023.pdf|pdf]])


* Copilot: File System ([[Media:glibcFileSystem.20251029-2.pdf|pdf]])
* Copilot: File Buffer ([[Media:glibcFileBuffer.20251025-2.pdf|pdf]])
* Copilot: File IO ([[Media:glibcFileIO.20251025-2.pdf|pdf]])
* Copilot: File Permission ([[Media:glibcFilePerm.20260121.pdf|pdf]])
* Copilot: File Control ([[Media:CP.FileCntl.20260428.pdf|pdf]], [[Media:CP.FileCntl.A.20260519.pdf|A]], [[Media:CP.FileCntl.B.20260504.pdf|B]], [[Media:CP.FileCntl.C.20260501.pdf|C]])

&lt;br&gt;
&lt;br&gt;

== Process ==
* Process  ([[Media:SysP.Process.20251120.pdf|pdf]])
* Fork  ([[Media:SysP.Fork.20251126.pdf|pdf]])


* Copilot: Process Information ([[Media:glibc.Process.1Info.20251101.pdf|pdf]])
* Copilot: Process Control ([[Media:glibc.Process.2Control.20251103.pdf|pdf]])
* Copilot: Process Execution ([[Media:glibc.Proc.3Exec.20251105.pdf|pdf]])
* Copilot: Process Fork ([[Media:glibc.Proc.4Fork.20251106.pdf|pdf]])
* Copilot: Process Context Switching ([[Media:glibc.Proc.5Context.20251107.pdf|pdf]])
* Copilot: Process Exec family of functions ([[Media:glibc.Proc.6ExecCall.20251112.pdf|pdf]])
* Copilot: Process Wait family of functions ([[Media:glibc.Proc.7WaitCall.20251112.pdf|pdf]])
* Copilot: Process Exit ([[Media:glibc.Proc.8Exit.20251113.pdf|pdf]])

&lt;/br&gt;

== Inter Process Communication==

=== Signal ===
* Signal  ([[Media:SysP.7.A.Signal.20121206.pdf|pdf]])


* Copilot: Signal 1. Alarm ([[Media:glibc.Signal.Alarm.20251201.pdf|pdf]])
* Copilot: Signal 2. Other Functions ([[Media:glibc.Signal.2Other.20251205.pdf|pdf]])

&lt;/br&gt;

=== Pipe ===

* Pipe ([[Media:SysP.3.A.IPC.20121115.pdf|pdf]])


* Copilot: Pipe 1. A Special File ([[Media:glibc.Pipe.File.20260307.pdf|pdf]])

&lt;/br&gt;

=== System V IPC ===

* Message Queue  ([[Media:SysP.5.A.MessageQ.20121213.pdf|pdf]])

* Shared Memory  ([[Media:SysP.8.A.SharedMem.20121227.pdf|pdf]])

* Semaphore  ([[Media:SysP.6.A.Semaphore.20251215.pdf|pdf]])

&lt;/br&gt;

* Copilot: Message Queue ([[Media:glibc.MessageQ.20251202.pdf|pdf]])
* Copilot: Shared Memory ([[Media:glibc.SharedMem.20251203.pdf|pdf]])
* Copilot: Semaphore ([[Media:glibc.Semaphore.20251215.pdf|pdf]])

&lt;/br&gt;

=== Socket ===

* Socket  ([[Media:SysP.4.A.Socket.20121122.pdf|pdf]])

&lt;/br&gt;

== Thread ==
* POSIX thread (pthread)  ([[Media:SysP.9.A.Pthread.20130225.pdf|pdf]])

==External links==
* [http://www.tldp.org/LDP/tlk/tlk.html The Linux Kernel]
* [http://www.tldp.org/LDP/lpg/lpg.html The Linux Programmer's Guide]
* [http://www.cs.cf.ac.uk/Dave/C/ Programming in C - UNIX System Calls and Subroutines using C.]
* [http://www.cs.cmu.edu/afs/cs/academic/class/15492-f07/www/pthreads.html POSIX thread (pthread) libraries]
* [https://computing.llnl.gov/tutorials/pthreads/#Thread POSIX Threads Programming]

[[Category:Linux]]
[[Category:Computer programming]]
[[Category:C programming language]]</text>
      <sha1>7s6kzcmg1egiubwxcjdpnjs813sz8kz</sha1>
    </revision>
    <revision>
      <id>2810560</id>
      <parentid>2810556</parentid>
      <timestamp>2026-05-20T07:26:58Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* File System */</comment>
      <origin>2810560</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3197" sha1="i0kuf6naiajo5zr4vxa99x3d7ppbwpr" xml:space="preserve">This course belongs to the [[Electrical &amp; Computer Engineering Studies]]

== Introduction ==
* Introduction ([[Media:SysP.Intro.20161128.pdf|pdf]])

== File System ==
* File System ([[Media:SysP.FileSystem.20251023.pdf|pdf]])
* File Pointer ([[Media:SysP..FilePointer.20161103.pdf|pdf]])
* System Calls ([[Media:SysP.File.SysCall.20161128.pdf|pdf]])
* File IO ([[Media:SysP.FileIO.20251023.pdf|pdf]])


* Copilot: File System ([[Media:glibcFileSystem.20251029-2.pdf|pdf]])
* Copilot: File Buffer ([[Media:glibcFileBuffer.20251025-2.pdf|pdf]])
* Copilot: File IO ([[Media:glibcFileIO.20251025-2.pdf|pdf]])
* Copilot: File Permission ([[Media:glibcFilePerm.20260121.pdf|pdf]])
* Copilot: File Control ([[Media:CP.FileCntl.20260428.pdf|pdf]], [[Media:CP.FileCntl.A.20260520.pdf|A]], [[Media:CP.FileCntl.B.20260504.pdf|B]], [[Media:CP.FileCntl.C.20260501.pdf|C]])

&lt;br&gt;
&lt;br&gt;

== Process ==
* Process  ([[Media:SysP.Process.20251120.pdf|pdf]])
* Fork  ([[Media:SysP.Fork.20251126.pdf|pdf]])


* Copilot: Process Information ([[Media:glibc.Process.1Info.20251101.pdf|pdf]])
* Copilot: Process Control ([[Media:glibc.Process.2Control.20251103.pdf|pdf]])
* Copilot: Process Execution ([[Media:glibc.Proc.3Exec.20251105.pdf|pdf]])
* Copilot: Process Fork ([[Media:glibc.Proc.4Fork.20251106.pdf|pdf]])
* Copilot: Process Context Switching ([[Media:glibc.Proc.5Context.20251107.pdf|pdf]])
* Copilot: Process Exec family of functions ([[Media:glibc.Proc.6ExecCall.20251112.pdf|pdf]])
* Copilot: Process Wait family of functions ([[Media:glibc.Proc.7WaitCall.20251112.pdf|pdf]])
* Copilot: Process Exit ([[Media:glibc.Proc.8Exit.20251113.pdf|pdf]])

&lt;/br&gt;

== Inter Process Communication==

=== Signal ===
* Signal  ([[Media:SysP.7.A.Signal.20121206.pdf|pdf]])


* Copilot: Signal 1. Alarm ([[Media:glibc.Signal.Alarm.20251201.pdf|pdf]])
* Copilot: Signal 2. Other Functions ([[Media:glibc.Signal.2Other.20251205.pdf|pdf]])

&lt;/br&gt;

=== Pipe ===

* Pipe ([[Media:SysP.3.A.IPC.20121115.pdf|pdf]])


* Copilot: Pipe 1. A Special File ([[Media:glibc.Pipe.File.20260307.pdf|pdf]])

&lt;/br&gt;

=== System V IPC ===

* Message Queue  ([[Media:SysP.5.A.MessageQ.20121213.pdf|pdf]])

* Shared Memory  ([[Media:SysP.8.A.SharedMem.20121227.pdf|pdf]])

* Semaphore  ([[Media:SysP.6.A.Semaphore.20251215.pdf|pdf]])

&lt;/br&gt;

* Copilot: Message Queue ([[Media:glibc.MessageQ.20251202.pdf|pdf]])
* Copilot: Shared Memory ([[Media:glibc.SharedMem.20251203.pdf|pdf]])
* Copilot: Semaphore ([[Media:glibc.Semaphore.20251215.pdf|pdf]])

&lt;/br&gt;

=== Socket ===

* Socket  ([[Media:SysP.4.A.Socket.20121122.pdf|pdf]])

&lt;/br&gt;

== Thread ==
* POSIX thread (pthread)  ([[Media:SysP.9.A.Pthread.20130225.pdf|pdf]])

==External links==
* [http://www.tldp.org/LDP/tlk/tlk.html The Linux Kernel]
* [http://www.tldp.org/LDP/lpg/lpg.html The Linux Programmer's Guide]
* [http://www.cs.cf.ac.uk/Dave/C/ Programming in C - UNIX System Calls and Subroutines using C.]
* [http://www.cs.cmu.edu/afs/cs/academic/class/15492-f07/www/pthreads.html POSIX thread (pthread) libraries]
* [https://computing.llnl.gov/tutorials/pthreads/#Thread POSIX Threads Programming]

[[Category:Linux]]
[[Category:Computer programming]]
[[Category:C programming language]]</text>
      <sha1>i0kuf6naiajo5zr4vxa99x3d7ppbwpr</sha1>
    </revision>
  </page>
  <page>
    <title>Understanding Arithmetic Circuits</title>
    <ns>0</ns>
    <id>139384</id>
    <revision>
      <id>2810433</id>
      <parentid>2810118</parentid>
      <timestamp>2026-05-19T14:04:53Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Adder */</comment>
      <origin>2810433</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3135" sha1="21czon2moy3d4iwtzwrqw111e4lp9km" xml:space="preserve">
== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.1.A.CLA.20260109.pdf|org]], [[Media:VLSI.Arith.2A.CLA.20260519.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260519.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]]||
|| || 
|}
&lt;/br&gt;
 
=== 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
&lt;/br&gt;

== Barrel Shifter ==
* Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]])
&lt;/br&gt;

'''Mux Based Barrel Shifter'''
* Analysis  ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) 
* Implementation 
&lt;/br&gt;

== Multiplier == 
=== Array Multipliers ===
* Analysis  ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]])
&lt;/br&gt;

=== 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]])
&lt;/br&gt;

=== Booth Multipliers ===
* [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]]
* Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]])
&lt;/br&gt;

== Divider == 
* Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])&lt;/br&gt;

&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
      <sha1>21czon2moy3d4iwtzwrqw111e4lp9km</sha1>
    </revision>
  </page>
  <page>
    <title>Pronunciation of 'S' before 'Y'</title>
    <ns>0</ns>
    <id>148665</id>
    <revision>
      <id>2810455</id>
      <parentid>1221351</parentid>
      <timestamp>2026-05-19T16:10:53Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>project box(es)</comment>
      <origin>2810455</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1488" sha1="7zgqy2rg295tbxiq75mmmle0q394z0p" xml:space="preserve">{{languages}}
Many people get confused with the pronunciation '''''s''''' when precedes '''''y'''''. But here's a good point. An example could be shown, as in ''this year''. Normally we pronounce it as /ðɪs jɪr/ (IPA), but this is actually pronounced as /ðɪʃjiə(r)/ (IPA).

==Pronunciation of ''S''==
The letter ''s'' has a sound of /s/ (IPA) but it also sounds as /z/ and /ʒ/ and sometimes /ʃ/. For examples, me'''ss''' /mes/, ro'''s'''e /roʊz/, vi'''s'''ion /ˈvɪʒn/, and ti'''ss'''ue /'tɪʃu:/.

==Pronunciation of ''Y''==
Usually the letter ''y'' has a pronunciation of /j/ (IPA; spelled [y]). This letter is one of the two semivowels (the other is w). Words with the letter ''y'' are, youth /juːθ/, yolk /joʊk/ etc.

==Pronunciation of ''S'' when precedes ''Y''==
As mentioned above, ''this year'' is pronounced as /ðɪʃjiə(r)/, other words such as ''pass you'', ''guess you'', ''miss you'' etc. are pronounced with the same method.

Likewise, if a word ends with the pronunciation of /z/ (e.g. ''a'''s''''', ''i'''s''''' etc.) and precedes a word starting with ''y'', the pronunciation of /z/ should turn into /ʒ/. For examples, ''as you'' /æʒjuː/, ''is your'' /ɪˈʒjɔːr/ etc.

It doesn't have to be only ''y'', but the same rule will follow if a word starts with the pronunciation of /j/. For example, ''his university'' /hɪˌʒjuːnɪˈvɜ:rsətɪ/, ''miss universe'' /mɪˈʃjuːnɪvɜː(r)s/.

[[Category:English language]]
[[Category:Pronunciation]]</text>
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  </page>
  <page>
    <title>Wireless power</title>
    <ns>0</ns>
    <id>162869</id>
    <revision>
      <id>2810411</id>
      <parentid>1567238</parentid>
      <timestamp>2026-05-19T12:37:09Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>project boxes</comment>
      <origin>2810411</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4428" sha1="mbyygaopbd16tk78zqtyifjzqmmask1" xml:space="preserve">{{engineering}}
[[File:Original_Tesla_Coil.png|thumb|Tesla wireless coil transmitter]]
[[File:Tesla Broadcast Tower 1904.jpeg|thumb|Wardenclyffe Tower Long Island]]
==Problem==
Tesla transmitted power wirelessly before Edison's distribution system of power existed. Why aren't we doing this today? What is all this zero point free energy stuff? It works because cell phone/tablets can be recharged wirelessly. Is there a conspiracy? Answer these questions.

==Conceive==
Duplicate wireless delivery of energy to drive a light bulb or charge a cell phone in a manner similar to existing shipping products.

Demonstrate wireless energy transfer, show it's flaws.
*[https://www.youtube.com/watch?v=cXhZvyGtMrk video of fluorescent lights under power lines]
*[https://www.youtube.com/watch?v=pyGZTiSeKjY video LED light bulb under the power lines]

Find local utility/county/state laws that reference free energy or illegal connections to the grid.

Build simple device that runs on free energy and explain how it works. 
*[https://www.youtube.com/watch?v=WooCJ3mye54 video of simple, easy to build]
*[https://www.youtube.com/watch?v=oMK0dEKWJdY video complex, 3D printed Atmospheric Motor]

==Design==
Find the [[w:Qi_(inductive_power_standard)|Qi]] standard and review it technically.

{| class=&quot;wikitable&quot;
|-
|{{collapse top|Existing designs}}
* [[w:Wireless_power| wikipedia summary]] ... Shrink existing theory to a sound byte
*[http://www.instructables.com/id/Wireless-Power-Transmission-Over-Short-Distances-U/ follow instructable] HCC ID badge scanner hack
{{collapse bottom}}

{{collapse top|History}}
*spark gap
*spark gap filters
*[[w:Alexanderson_alternator|Alexanderson alternator]] ... [http://www.youtube.com/watch?v=neJeIxaMqF0 video]
*vaccuum tube
{{collapse bottom}}

{{collapse top|Zero-Point, Free Energy Hoaxes}}
* [[w:William_Reich|William Reich]] and [[w:Orgone|Orgone energy accumulator]]
* [http://www.thelivingmoon.com/41pegasus/02files/Zero_Point.html summary of the characters]
{{collapse bottom}}

{{collapse top|Design Starting points in Lab}}
* Using microwave fan connected to AC as source .. AC safety
* [[Wikipedia:Hawkins Electrical Guide#Google Books Volume Links|Hawkins]] [[w:Hawkins_Electrical_Guide|Electrical Guide]]
{{collapse bottom}}

{{collapse top|Energy from hot and cold water}}
[https://www.youtube.com/watch?v=RC16MwzFq8A&amp;list=UUSY6p1ZwMs0lW2XP7Zc7k9g Video] using [[wikipedia:Thermoelectric cooling|Peltier Module]] and the Seebeck Effect.
{{collapse bottom}}

{{collapse top|Motor running off static near field}}
[https://www.youtube.com/watch?v=ksp_O_1WmvA video]]
{{collapse bottom}}
|}

==Implement==
&lt;gallery&gt;
File:Wikiversity - Wireless Electricity Design project - First idea, with one turn of coil.jpg|First idea, with one turn 
File:Wikiversity - Wireless Electricity Design project - Second try with different coils.jpg|Second try with different materials
File:Wikiversity - Wireless Electricity Design project - Third try with a large coil.jpg|Third try with a large coil
File:Magnetic field checker.JPG|Magnetic field our coil is producing
&lt;/gallery&gt;

==Operate==
Develop a demon that others can use to talk about this in a variety of settings.

==Demo==
*video
*[https://docs.google.com/presentation/d/1c8uGSBVeBCkQ7r2rWEjiFEr5v6TbVNHjua8t5TcJdg8/edit?usp=sharing presentation]

==Next Steps==
* Reverse engineer and document device made from disk eraser
* purchase some Qi devices and reverse engineer
* [[w:Wireless_power| wikipedia summary]] ... Shrink existing theory to a sound byte
*[http://www.instructables.com/id/Wireless-Power-Transmission-Over-Short-Distances-U/ follow instructable] HCC ID badge scanner hack
*build free energy device from some youtube or other internet site and explain how it works/doesn't work
*put together a history of far field transmission
**spark gap
**spark gap filters
**[[w:Alexanderson_alternator|Alexanderson alternator]] ... [http://www.youtube.com/watch?v=neJeIxaMqF0 video]
**vaccuum tube
*estimate local free energy available from radio and tv stations in local area using FCC maps and station transmitter sizes
*go through Youtube free energy video's and estimate power received
* Summarize High-Voltage [http://www.bpa.gov/news/pubs/GeneralPublications/lusi-Living-and-working-safely-around-high-voltage-power-lines.pdf Power Line Saftey]

[[Category:Active General Engineering Projects]]
[[Category:Static electricity]]</text>
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  </page>
  <page>
    <title>Complex analysis in plain view</title>
    <ns>0</ns>
    <id>171005</id>
    <revision>
      <id>2810440</id>
      <parentid>2809649</parentid>
      <timestamp>2026-05-19T14:25:17Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Geometric Series Examples */</comment>
      <origin>2810440</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5296" sha1="dolq9c8ul5i4ylhf4lb7z0szcutdz2k" xml:space="preserve">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 &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, 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 &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;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.20260518.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]]</text>
      <sha1>dolq9c8ul5i4ylhf4lb7z0szcutdz2k</sha1>
    </revision>
    <revision>
      <id>2810443</id>
      <parentid>2810440</parentid>
      <timestamp>2026-05-19T14:29:04Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Geometric Series Examples */</comment>
      <origin>2810443</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5296" sha1="at190kd7yz54dxd5t4po1bzph88t59q" xml:space="preserve">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 &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, 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 &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;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.20260519.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]]</text>
      <sha1>at190kd7yz54dxd5t4po1bzph88t59q</sha1>
    </revision>
  </page>
  <page>
    <title>The necessities in Filter Theory</title>
    <ns>0</ns>
    <id>199550</id>
    <revision>
      <id>2810562</id>
      <parentid>2809788</parentid>
      <timestamp>2026-05-20T08:08:27Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Sample Processing Methods */</comment>
      <origin>2810562</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1524" sha1="1op3njzlnnbns4c7p5zmbq61cn9w15s" xml:space="preserve">==''' Background '''==
=== Bode plot ===

See [http://lpsa.swarthmore.edu/Bode/Bode.html swarthmore]
&lt;/br&gt;

=== OP Amp ===
Overview ([[Media:OPAmp.A.1.20151203.pdf  |pdf]])
 
See [http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/opampcon.html#c1 Hyperphysics]
&lt;/br&gt;

==''' Analog Filter Analysis (Continuous Time) '''==
=== First Order Filters ===
&lt;/br&gt;

=== Second Order Filters ===
&lt;/br&gt;

==''' Digital Filter Analysis (Discrete Time) '''==

=== Sample Processing Methods  ===
* Tapped Delays ([[Media:Sample.TappedDelay.20260518.pdf  |A.pdf]])
* Programming Considerations
* Circular Buffers

=== FIR Filter Realizations ===
* Direct Form FIR Filter 
* Canonical Form FIR Filter
* Cascade Form FIR Filter

=== IIR Filter Realizations ===
* Direct Form IIR Filter ([[Media:IIR.DirectForm.20231209.pdf  |A.pdf]])
* Canonical Form IIR Filter
* Cascade Form IIR Filter 
&lt;/br&gt;

=== FIR (Finite Impulse Response) Filters ===
* Block Processing Methods
* Sample Processing Methods
* Window Method
* Kaiser Window
* Frequency Sampling Method
&lt;/br&gt;

=== IIR (Infinite Impulse Response) Filters ===
* Bilinear Transform
* 1st Order Lowpass and Highpass Filters
* 2nd Order Lowpass and Highpass Filters
* Parametric Equalizer Filters
* Comb Filters
* High Order Filters
&lt;/br&gt;


=== Example Octave Codes for Digital Filters === 

==== Octave Functions for Filters ====
* Octave Functions for Filters ([[Media:Octave.1.Function.1.A.20180219.pdf  |A.pdf]]) 


&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>1op3njzlnnbns4c7p5zmbq61cn9w15s</sha1>
    </revision>
    <revision>
      <id>2810564</id>
      <parentid>2810562</parentid>
      <timestamp>2026-05-20T08:09:50Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Sample Processing Methods */</comment>
      <origin>2810564</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1524" sha1="peyiuetnbg1atm7koou5ga5bd92v4if" xml:space="preserve">==''' Background '''==
=== Bode plot ===

See [http://lpsa.swarthmore.edu/Bode/Bode.html swarthmore]
&lt;/br&gt;

=== OP Amp ===
Overview ([[Media:OPAmp.A.1.20151203.pdf  |pdf]])
 
See [http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/opampcon.html#c1 Hyperphysics]
&lt;/br&gt;

==''' Analog Filter Analysis (Continuous Time) '''==
=== First Order Filters ===
&lt;/br&gt;

=== Second Order Filters ===
&lt;/br&gt;

==''' Digital Filter Analysis (Discrete Time) '''==

=== Sample Processing Methods  ===
* Tapped Delays ([[Media:Sample.TappedDelay.20260519.pdf  |A.pdf]])
* Programming Considerations
* Circular Buffers

=== FIR Filter Realizations ===
* Direct Form FIR Filter 
* Canonical Form FIR Filter
* Cascade Form FIR Filter

=== IIR Filter Realizations ===
* Direct Form IIR Filter ([[Media:IIR.DirectForm.20231209.pdf  |A.pdf]])
* Canonical Form IIR Filter
* Cascade Form IIR Filter 
&lt;/br&gt;

=== FIR (Finite Impulse Response) Filters ===
* Block Processing Methods
* Sample Processing Methods
* Window Method
* Kaiser Window
* Frequency Sampling Method
&lt;/br&gt;

=== IIR (Infinite Impulse Response) Filters ===
* Bilinear Transform
* 1st Order Lowpass and Highpass Filters
* 2nd Order Lowpass and Highpass Filters
* Parametric Equalizer Filters
* Comb Filters
* High Order Filters
&lt;/br&gt;


=== Example Octave Codes for Digital Filters === 

==== Octave Functions for Filters ====
* Octave Functions for Filters ([[Media:Octave.1.Function.1.A.20180219.pdf  |A.pdf]]) 


&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>peyiuetnbg1atm7koou5ga5bd92v4if</sha1>
    </revision>
    <revision>
      <id>2810566</id>
      <parentid>2810564</parentid>
      <timestamp>2026-05-20T08:10:53Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Sample Processing Methods */</comment>
      <origin>2810566</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1524" sha1="3g0lora0fgnqaref159sl86vpbid2v0" xml:space="preserve">==''' Background '''==
=== Bode plot ===

See [http://lpsa.swarthmore.edu/Bode/Bode.html swarthmore]
&lt;/br&gt;

=== OP Amp ===
Overview ([[Media:OPAmp.A.1.20151203.pdf  |pdf]])
 
See [http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/opampcon.html#c1 Hyperphysics]
&lt;/br&gt;

==''' Analog Filter Analysis (Continuous Time) '''==
=== First Order Filters ===
&lt;/br&gt;

=== Second Order Filters ===
&lt;/br&gt;

==''' Digital Filter Analysis (Discrete Time) '''==

=== Sample Processing Methods  ===
* Tapped Delays ([[Media:Sample.TappedDelay.20260520.pdf  |A.pdf]])
* Programming Considerations
* Circular Buffers

=== FIR Filter Realizations ===
* Direct Form FIR Filter 
* Canonical Form FIR Filter
* Cascade Form FIR Filter

=== IIR Filter Realizations ===
* Direct Form IIR Filter ([[Media:IIR.DirectForm.20231209.pdf  |A.pdf]])
* Canonical Form IIR Filter
* Cascade Form IIR Filter 
&lt;/br&gt;

=== FIR (Finite Impulse Response) Filters ===
* Block Processing Methods
* Sample Processing Methods
* Window Method
* Kaiser Window
* Frequency Sampling Method
&lt;/br&gt;

=== IIR (Infinite Impulse Response) Filters ===
* Bilinear Transform
* 1st Order Lowpass and Highpass Filters
* 2nd Order Lowpass and Highpass Filters
* Parametric Equalizer Filters
* Comb Filters
* High Order Filters
&lt;/br&gt;


=== Example Octave Codes for Digital Filters === 

==== Octave Functions for Filters ====
* Octave Functions for Filters ([[Media:Octave.1.Function.1.A.20180219.pdf  |A.pdf]]) 


&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>3g0lora0fgnqaref159sl86vpbid2v0</sha1>
    </revision>
  </page>
  <page>
    <title>Haskell programming in plain view</title>
    <ns>0</ns>
    <id>203942</id>
    <revision>
      <id>2810464</id>
      <parentid>2810131</parentid>
      <timestamp>2026-05-19T17:02:44Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Lambda Calculus */</comment>
      <origin>2810464</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6273" sha1="t5ovdof610q6cndvi4hspr3egpnbf9z" xml:space="preserve">
==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]])
&lt;/br&gt;

==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]])

&lt;/br&gt;

==Using GHCi==
* Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]])
&lt;/br&gt;

==Using Libraries==
* Library ([[Media:Library.1.A.20170605.pdf |pdf]])
&lt;/br&gt;
&lt;/br&gt;


==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]])
&lt;/br&gt;
&lt;/br&gt;

==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.20260519.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]])
&lt;/br&gt;
&lt;/br&gt;

==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 &amp; 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 &amp; 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]])
&lt;/br&gt;

==Polymorphism==
* Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]])
&lt;/br&gt;

==Concurrent Haskell ==
&lt;/br&gt;


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]]</text>
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  <page>
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        <id>21186</id>
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      <comment>/* Using Libraries */</comment>
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      <text bytes="2647" sha1="8nx8xd4hzy147z1m0k25aqr8bncb665" xml:space="preserve">==''' Part I '''==
&lt;!----------------------------------------------------------------------&gt;
===  Introduction ===

* Overview 
* Memory 
* Number 

&lt;!----------------------------------------------------------------------&gt;


=== Python for C programmers ===

* Hello, World! ([[Media:CProg.Hello.1A.20230406.pdf |pdf]])
* Statement Level  ([[Media:CProg.Statement.1A.20230509.pdf |pdf]])
* Output with print
* Formatted output
* File IO

&lt;!----------------------------------------------------------------------&gt;


=== Using Libraries ===

* Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]])
* Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]])
* Packages ([[Media:Python.Work2.Package.1A.20241207.pdf |pdf]])
* Libraries ([[Media:Python.Work2.Library.1A.20260519.pdf |pdf]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

===   Handling Repetition ===
 
* Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]])

* Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling a Big Work ===

* Functions ([[Media:Python.Work1.Function.1A.20230529.pdf |pdf]])
* Lambda ([[Media:Python.Work2.Lambda.1A.20230705.pdf |pdf]])
* Type Annotations ([[Media:Python.Work2.AtypeAnnot.1A.20230817.pdf |pdf]])

  
&lt;!----------------------------------------------------------------------&gt;

=== Handling Series of Data ===

* Arrays ([[Media:Python.Series1.Array.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series2.Tuple.1A.pdf |pdf]])
* Lists ([[Media:Python.Series3.List.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series4.Tuple.1A.pdf |pdf]])
* Sets ([[Media:Python.Series5.Set.1A.pdf |pdf]])
* Dictionary ([[Media:Python.Series6.Dictionary.1A.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling Various Kinds of Data ===
  
* Types 
* Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]])
* Files ([[Media:Python.Data4.File.1.A.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

=== Class and Objects ===
  
* Classes &amp; Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]])
* Inheritance

 
&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;

== Python in Numerical Analysis  ==
 
&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
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  <page>
    <title>NCERT/Textbook Solutions/Class VII/Mathematics</title>
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      <contributor>
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        <id>276019</id>
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      <minor />
      <comment>Reverted edit by [[Special:Contributions/Arjun.paraheights|Arjun.paraheights]] ([[User_talk:Arjun.paraheights|talk]]) to last version by [[User:MathXplore|MathXplore]] using [[Wikiversity:Rollback|rollback]]</comment>
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      <text bytes="1434" sha1="rwwsue818cd9wwvmuehqrb9g7vwi9tj" xml:space="preserve">[[Wikipedia: National Council of Educational Research and Training|'''NCERT''']] books are based upon the curriculum/syllabus defined by [[Wikipedia: Central Board of Secondary Education|CBSE]]. These syllabus are periodically reviewed and revised. The NCERT book for a particular subject is divided into various chapters and every chapter has a set of questions following the chapter. This section provides answers to the questions at the end of each chapter in the '''Maths''' book, for  [[NCERT/Textbook_Solutions#Solutions_to_Textbooks_of_Class_VII|'''Class-VII''']].

==Chapter 01 Integers==

==Chapter 02 Fractions and Decimals==

==Chapter 03 Data Handling==

==Chapter 04 Simple Equations==

==Chapter 05 Lines and Angles==

==Chapter 06 Triangles and its properties==

==Chapter 07 Congruence of Triangles==

==Chapter 08 Comparing Quantities==

==Chapter 09 Rational Numbers==

==Chapter 10 Practical Geometry==

==Chapter 11 Perimeter and Area==

==Chapter 12 Algebraic Expressions==

==Chapter 13 Exponents and Powers==

==Chapter 14 Symmetry==

==Chapter 15 Visualising Solid Shapes==

==See Also==
* [[NCERT/Textbook_Solutions]]
* [[NCERT/Textbook_Solutions#Solutions_to_Textbooks_of_Class_VII]]

==External Links==
* [http://www.ncert.nic.in/index.html Official website of NCERT]
* [http://epathshala.nic.in/e-pathshala-4/flipbook/ NCERT textbooks available online]

==References==
{{Reflist}}

[[Category:Mathematics]]</text>
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      <comment>/* */ Welcome to Gboard clipboard, any text you copy will be saved here.Use the edit icon to pin, add or delete clips.Touch and hold a clip to pin it. Unpinned clips will be deleted after 1 hour.Tap on a clip to paste it in the text box.</comment>
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      <text bytes="2665" sha1="f2n4sqc2jex4yph0ultfe2pxh3hy9rz" xml:space="preserve">== Courses ==
* [[Python Concepts]]
* [[Python Programming]]
* [[Python/Serial port/pySerial|Serial communication with pyserial]]{{stage short|100%}}
* [[Python/MQTT|MQTT client with paho-mqtt library]]{{stage short|100%}}
* All Python ''{{Subpages/Simple}}''

== Examples ==
* [[Python/Time, Distance, and Speed|Time, Distance, and Speed]]
* [[/Prime factorization/]]
* [[/Musical intervals (numpy matplotlib)/]]
* [[Handler for references at Wikiversity pages]]
* [[Python Programming/GUI/Serial monitor|Develop the Arduino serial monitor-like with Tkinter and 3rd libary pySerial]]{{stage short|100%}}
* [[Python Programming/GUI/Oscilloscope|Develop the Oscilloscope-like desktop application with Tkinter, Matplotlib and 3rd libary pySerial]]{{stage short|100%}}

== Resources ==
* [[/pip (package manager)/]]

== Multimedia ==
* [https://www.youtube.com/watch?v=Y8Tko2YC5hA YouTube: What is Python and Why You Must Learn It]
* [https://www.youtube.com/watch?v=kLZuut1fYzQ YouTube: What Can You Do with Python? - The 3 Main Applications]
* [https://www.youtube.com/watch?v=rfscVS0vtbw YouTube: Learn Python - Full Course for Beginners]
* [https://www.youtube.com/watch?v=_uQrJ0TkZlc YouTube: Python Tutorial for Beginners]
*[https://www.youtube.com/watch?v=Khc5jR9EGGg YouTube: Python Course - Learn Python]
*[https://cs50.harvard.edu/python/2022/ CS50's Python Course]

== Other Information ==
Python is a multi-paradigm programming language, that is dynamically typed and garbage-collected. Many of the capabilities that the Python language supports are object-oriented programming and functional programming. This language follows a philosophy, which consists of phrases such as:

* &quot;Beautiful is better than ugly&quot;
* &quot;Simple is better than Complex&quot; 
* &quot;Readability counts&quot;
* &quot;Explicit is better than implicit&quot;
* &quot;Complex is better than complicated&quot;

See [https://en.wikipedia.org/wiki/Zen_of_Python Zen of Python] for more information about this philosophy. 

Python aims for simplicity and a less-cluttered syntax, while allowing developers to have options for their preferred coding method. Python has many versions out for developers to use. This consists of Python 2 (now on Sunset Status) and Python 3.13 (October 2024).

== Also See ==
* [[Computer Programming]]
* [[Pyjamas]] port of Google Web Toolkit (GWT)
* [[Wikipedia: Python (programming language)]]
* [[Wikibooks: Python Programming]]
* [https://programiz.pro/learn/master-python Beginner Python Course]  
* [https://www.wscubetech.com/resources/python Python Tutorial] 
* [https://www.wscubetech.com/resources/python/compiler Python Compiler]

== References ==
{{Reflist}}
[[Category:Python| ]]</text>
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      <minor />
      <comment>Reverted edit by [[Special:Contributions/~2026-30209-91|~2026-30209-91]] ([[User_talk:~2026-30209-91|talk]]) to last version by [[User:Anonymous Agent|Anonymous Agent]] using [[Wikiversity:Rollback|rollback]]</comment>
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      <text bytes="3470" sha1="04179rq8skxu10uu30rpmstzc5a4kkk" xml:space="preserve">[[File:Python.svg|right|180 px|Python logo]]

'''Python''' is a widely used high-level&lt;ref&gt;Programming languages can be low-level or high-level. 

High-level languages can be more readable to humans, 

while low-level languages are harder to understand. Low-level languages are closer to machine code. High-level languages are closer to the English language &lt;/ref&gt;, general-purpose, interpreted&lt;ref&gt;There are interpreted and compiled programming languages: 

compiled languages output and executable file, while

interpreted languages are executed line-by-line, using the interpreter.&lt;/ref&gt;, dynamic programming language. Its design philosophy emphasizes code readability, and its syntax allows programmers to express concepts in fewer lines of code than possible in other popular programming languages.

== Courses ==
* [[Python Concepts]]
* [[Python Programming]]
* [[Python/Serial port/pySerial|Serial communication with pyserial]]{{stage short|100%}}
* [[Python/MQTT|MQTT client with paho-mqtt library]]{{stage short|100%}}
* All Python ''{{Subpages/Simple}}''

== Examples ==
* [[Python/Time, Distance, and Speed|Time, Distance, and Speed]]
* [[/Prime factorization/]]
* [[/Musical intervals (numpy matplotlib)/]]
* [[Handler for references at Wikiversity pages]]
* [[Python Programming/GUI/Serial monitor|Develop the Arduino serial monitor-like with Tkinter and 3rd libary pySerial]]{{stage short|100%}}
* [[Python Programming/GUI/Oscilloscope|Develop the Oscilloscope-like desktop application with Tkinter, Matplotlib and 3rd libary pySerial]]{{stage short|100%}}

== Resources ==
* [[/pip (package manager)/]]

== Multimedia ==
* [https://www.youtube.com/watch?v=Y8Tko2YC5hA YouTube: What is Python and Why You Must Learn It]
* [https://www.youtube.com/watch?v=kLZuut1fYzQ YouTube: What Can You Do with Python? - The 3 Main Applications]
* [https://www.youtube.com/watch?v=rfscVS0vtbw YouTube: Learn Python - Full Course for Beginners]
* [https://www.youtube.com/watch?v=_uQrJ0TkZlc YouTube: Python Tutorial for Beginners]
*[https://www.youtube.com/watch?v=Khc5jR9EGGg YouTube: Python Course - Learn Python]
*[https://cs50.harvard.edu/python/2022/ CS50's Python Course]

== Other Information ==
Python is a multi-paradigm programming language, that is dynamically typed and garbage-collected. Many of the capabilities that the Python language supports are object-oriented programming and functional programming. This language follows a philosophy, which consists of phrases such as:

* &quot;Beautiful is better than ugly&quot;
* &quot;Simple is better than Complex&quot; 
* &quot;Readability counts&quot;
* &quot;Explicit is better than implicit&quot;
* &quot;Complex is better than complicated&quot;

See [https://en.wikipedia.org/wiki/Zen_of_Python Zen of Python] for more information about this philosophy. 

Python aims for simplicity and a less-cluttered syntax, while allowing developers to have options for their preferred coding method. Python has many versions out for developers to use. This consists of Python 2 (now on Sunset Status) and Python 3.13 (October 2024).

== Also See ==
* [[Computer Programming]]
* [[Pyjamas]] port of Google Web Toolkit (GWT)
* [[Wikipedia: Python (programming language)]]
* [[Wikibooks: Python Programming]]
* [https://programiz.pro/learn/master-python Beginner Python Course]  
* [https://www.wscubetech.com/resources/python Python Tutorial] 
* [https://www.wscubetech.com/resources/python/compiler Python Compiler]

== References ==
{{Reflist}}
[[Category:Python| ]]</text>
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{{secondary}}
[[File:Code of Hammurabi IMG 1936.JPG|thumb|Code of Hammurabi]]
Of the several law codes surviving from the ancient Middle East, the most famous after the '''Torah''' is the '''Code of Hammurabi'''. This copy was made long after Hammurabi’s time, and it is clear that his was a '''long-lasting contribution to Mesopotamian civilization'''. It encodes many laws that had probably evolved over a long period of time, but is interesting to the general reader because of what it tells us about the attitudes and daily lives of the ancient Babylonians. In the following selection, most of the long prologue praising Hammurabi’s power and wisdom is omitted.
==Laws==
15: If anyone take a male or female slave of the court, or a male/female slave of a freed man, outside the city gates [to escape; to be free without permission; to be free without any approval; to be free without any verdict], he shall be put to death [he/she shall die; execution].&lt;br&gt;
16: If any one receive into his house a runaway male/female slave of the court [If anyone keeps/holds an illegal slave], or of a freedman [slave of a free-man], and does not bring it out at the public proclamation [Referring to if doesn’t bring out after requests] of the [police] (Governing body that is similar to present-day policemen and policewomen), the master of the house shall be put to death [he/she shall die; execution].&lt;br&gt;
53: If any one be lazy to keep his dam in proper condition [If one does not maintain their dam], and does not so keep it; if then the dam breaks and all the fields be flooded [the dam breaks and the fields (crops) are flooded/drowned down], then shall he in whose dam the break occurred [owner of the dam] be sold for money, and the money shall replace the [grain] which he has caused to be ruined [ruined crops due to the drowning/flooding]&lt;br&gt;
54: If he be not able to replace the [grain] [replace the flooded grains], then he and his possessions shall be divided among the farmers whose he has flooded. [“Corn” possibly refers to the flooded grains].&lt;br&gt;
108: If a female [who sells wine; wine-seller] does not accept [grain] according to gross weight in payment of drink [salary in accordance to how much the wine is being sold/has been sold for], but takes money, and the price of the drink is less than that of the corn [“corn” might refer to salary/money, or might pertain to the actual food], she shall be convicted and thrown into the water [drowned].&lt;br&gt;
109: If conspirators meet in the house of a female [who sells wine; wine-seller], and these conspirators are not captured and delivered to the court [of justice/prosecution], the female wine-seller shall be put to death [execution, or the like...].&lt;br&gt;
110: If a sister of a nun open[s] a tavern to drink, then shall this woman [then this woman] be burned to death [lit on fire].&lt;br&gt;
138: If a man wishes to separate from his wife [divorce] who has borne him no children, he shall give her the amount of her purchase money and the dowry which she brought from her father's house, and let her go.&lt;br&gt;
141: If a man's wife, who lives in his house, wishes to leave it, plunges into debt [to go into business], tries to ruin her house, neglects her husband, and is judicially convicted: if her husband offer her release, she may go her way, and he gives her nothing as a gift of release. If her husband does not wish to release her, and if he take another wife, she shall remain as servant in her husband's house.&lt;br&gt;
142: If a woman quarrel with her husband, and say: &quot;You are not congenial to me&quot;, the reasons for her prejudice must be presented. If she is guiltless, and there is no fault on her part, but he leaves and neglects her, then no guilt attaches to this woman, she shall take her dowry and go back to her father's house.&lt;br&gt;
143: If she is not innocent, but leaves her husband, and ruins her house, neglecting her husband, this woman shall be cast[ed] into the water [drowned].&lt;br&gt;
198: If he put out the eye of a [commoner], or break the bone of a [commoner], he shall pay one [silver] mina.&lt;br&gt;
==Extra links==
{{wikipedia}}

[[Category:Mesopotamia]]</text>
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==Watersheds==
[[File:Continental divide on Loveland pass 3654 m.jpg|thumb|right|The Continental Divide in Colorado]]
River systems are divided into regions called '''watersheds''', which is the area of land that is drained by a water system. The largest watershed in the US is the Mississippi River watershed, which has hundreds of '''tributaries''' (a stream that flows into a lake/larger stream) that extend from the Rocky Mountains, in the West, to the Appalachian Mountains, and in the East.

There are several watersheds in the US, major and minor. Watersheds are separated from each other by an area of higher ground, known as the '''divide'''. In the US, the divide [in the US] is known as the ''Continental Divide''.

==Stream Erosion==
As a stream forms, it erodes soil/rock to make a '''channel''', which is the path that a stream follows. When a stream forms, its channel is usually thin and steep. But over time, the stream carries rock and soil downstream, and makes the channel a lot wider and deeper. When streams become longer and wider, this is what we call a '''river'''. A stream's ability to erode is influenced by three factors: gradient, discharge, and load.
===Gradient===
'''Gradient''' is the measure of the changes in elevation over a specific distance. A high gradient allows a stream to have more erosive energy to erode rock and soil, while the opposite has less erosive energy. 
===Discharge===
The amount of water that a stream/river contain in a given amount of time is called '''discharge'''. The discharge of a stream increases when a major storm occurs, or when hot weather quickly melts snow. As the stream's discharge increases, its erosive energy, speed, and the amount of materials that the stream can [be able to hold] increases as well.
===Load===
Materials carried by a stream are called the stream's '''load'''. The size of load = [affected] = stream's speed. Fast moving streams can carry a '''bed load''' (large materials, such as pebbles and boulders), while slow-moving stream can carry a '''suspended load''' (small rocks and soil in suspension). The '''dissolved load''' is material carried in solution (material is dissolved in the water). Some of the materials in the dissolved load are sodium and calcium.

==Stages of a River==
===Youthful River===
*Erodes its channel deeper than wider
*Flows quickly due to the steep gradient
*Channel is narrow and straight
*Tumbles over rocks in rapids and waterfalls
===Mature River===
*Erodes its channel wider than deeper
*Not a steep gradient
*Fewer falls and rapids
*Fed by many tributaries
*Has more discharge (amount of water that a stream/river carries in a specific amount of time)
===Old River===
*Low gradient
*Little erosive energy
*Deposits rock and soil in/along its channel than widen and deepen its banks
*Characterized by wide, flat &quot;flood plains&quot; (valleys), and many bends.
*Fewer tributaries than a mature river. 
===Rejuvenated Rivers===
*These rivers are found where the land is raised by tectonic activity. 
*Land rises: Gradient becomes steeper
*Land rises: River flows more quickly
*Land rises: Due to the gradient, river cuts more deeply into the valley floor
*Terraces form, which are step-like formations that often form on both sides of a stream valley as a result of rejuvenation.

{{DEFAULTSORT:Flow Of Water}}
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{{engineering}}
Zinc Finger Nucleases (ZFNs), Transcription Like Effector Nucleases (TALENs), Clustered Regularly Interspaced Short Palidromic Repeats (CRISPR/CRISPR- Cas) are the useful tools of genome editing. They function via utilizing target- specific nucleases &lt;ref&gt;Urnov, F. D., et al. (2010). &quot;Genome editing with engineered zinc finger nucleases.&quot; Nature Reviews Genetics 11(9): 636.&lt;/ref&gt;&lt;ref&gt;Joung, J. K. and J. D. Sander (2013). &quot;TALENs: a widely applicable technology for targeted genome editing.&quot; Nature reviews Molecular cell biology 14(1): 49.&lt;/ref&gt;&lt;ref&gt;Mali, P., et al. (2013). &quot;Cas9 as a versatile tool for engineering biology.&quot; Nature methods 10(10): 957.&lt;/ref&gt;.

Two main components of these tools are the targetable nucleases in order to create Double strand Breaks (DSBs) and a DSB repair mechanism to reconnect separated DNA parts, with the addition exogeneous DNA fragment. 

Two options for DNA repairing are Non-homologous End Joining (NHEJ) and Homology Directed Repair (HDR). Exogeneous genes connected by means of NHEJ repair generally gain deletion or/and insertion mutations. NHEJ is known to be error-prone as HDR is not, however HDR needs a repair template. This problem of HDR is overcome by co-introduction of exogeneous repair donors with target specific nucleases &lt;ref&gt;Ochiai, H. and T. Yamamoto (2015). Genome editing using zinc-finger nucleases (ZFNs) and transcription activator-like effector nucleases (TALENs). Targeted Genome Editing Using Site-Specific Nucleases, Springer: 3-24.&lt;/ref&gt;. NHEJ repair system functions regarless of phases of cell cycle in opposition to HDR which majorly works at S/G2 phase. 

Because DSB repair machinaries can be encouraged in most of the organisms, target specific nuclease- mediated genome engineering is able to be applied in animals and plants &lt;ref&gt;Wolfe, S. A., et al. (2000). &quot;DNA recognition by Cys2His2 zinc finger proteins.&quot; Annual review of biophysics and biomolecular structure 29(1): 183-212.&lt;/ref&gt;. 

==  ZFNs  ==
ZFNs are artificial structures that are the combination of a restriction enzyme FokI domain and  a C2H2 zinc-finger (ZF) DNA-binding domain &lt;ref&gt;Wolfe, S. A., et al. (2000). &quot;DNA recognition by Cys2His2 zinc finger proteins.&quot; Annual review of biophysics and biomolecular structure 29(1): 183-212.&lt;/ref&gt;.

In past, scientist observed the ability of ZF domain to interract and bind to a variety of nucleotide sequence so they engineered this domain for tagetting user defined site in the genome &lt;ref&gt;Desjarlais, J. R. and J. M. Berg (1992). &quot;Redesigning the DNA‐binding specificity of a zinc finger protein: A data base‐guided approach.&quot; Proteins: Structure, Function, and Bioinformatics 12(2): 101-104.&lt;/ref&gt;.

Main characteristics of ZF domain originate from a zinc ion coordinated by two cystein residue, located on two antiparallel β-sheets, and two histidine residues on the α-helix. In nature, the ZF domain is primarily responsible for connection with DNA, each finger is capable of detecting  a 3 nucleotide unit, called target triplet, by means of its α- helix, thus named also recognition helix. Linkage to DNA only occurs at single strand. A 3 finger ZF can detect 9 base, at total. Furthermore, ZFNs basically works as dimers so recognition capacity at total is 18 base. In human genome, 18  base is a satisfactory length to specifically target a single site on the genome &lt;ref&gt;Wolfe, S. A., et al. (2000). &quot;DNA recognition by Cys2His2 zinc finger proteins.&quot; Annual review of biophysics and biomolecular structure 29(1): 183-212.&lt;/ref&gt;.

It is possible to observe unintended interractions between some parts of ZFNs and DNA. Another problem is that each finger strongly prefer to connect to GNN triplet. These issues harden design and limits target site repertoire for linkage of ZFN. 

FokI nuclease cuts target DNA away from the recognition location &lt;ref&gt;Li, L., et al. (1992). &quot;Functional domains in Fok I restriction endonuclease.&quot; Proceedings of the National Academy of Sciences 89(10): 4275-4279.&lt;/ref&gt;. FokI work as a dimer, it is an obligation for functioning of the domain. Firstly one of the FokI binds and then second FokI freely travelling in the solution connects to just directly opposite direction of the first one, thus FokI become dimerized. After that the dimer cut DNA &lt;ref&gt;Pernstich, C. and S. E. Halford (2011). &quot;Illuminating the reaction pathway of the FokI restriction endonuclease by fluorescence resonance energy transfer.&quot; Nucleic acids research 40(3): 1203-1213.&lt;/ref&gt;. 

Nuclease of ZFN solely efficiently dimerize and cleave DNA when two of ZFNs link to target nucleotides that are separated via the spacer sequence. Therefore ZFNs obtain high specifity based on this property &lt;ref&gt;Smith, J., et al. (2000). &quot;Requirements for double-strand cleavage by chimeric restriction enzymes with zinc finger DNA-recognition domains.&quot; Nucleic acids research 28(17): 3361-3369.&lt;/ref&gt;. However, it is probable to get off target effects, like cleavage of palidromic sequence. In order to escape from this type of unintended effects, obligate heterodimeric mutants of FokI restriction enzyme can be used. By the prevention of homodimerization of mutant nucleases, decline in cytotoxicity is reached &lt;ref&gt;Miller, J. C., et al. (2007). &quot;An improved zinc-finger nuclease architecture for highly specific genome editing.&quot; Nature biotechnology 25(7): 778.&lt;/ref&gt;&lt;ref&gt;Szczepek, M., et al. (2007). &quot;Structure-based redesign of the dimerization interface reduces the toxicity of zinc-finger nucleases.&quot; Nature biotechnology 25(7): 786.&lt;/ref&gt;&lt;ref&gt;Doyon, Y., et al. (2011). &quot;Enhancing zinc-finger-nuclease activity with improved obligate heterodimeric architectures.&quot; Nature methods 8(1): 74.&lt;/ref&gt;.
 
Wild type/nuclease-dead mutant FokI heterodimer cuts only single strand of DNA. As NHEJ repair is not able to recover this type of break, HDR takes over the repair work &lt;ref&gt;Kim, E., et al. (2012). &quot;Precision genome engineering with programmable DNA-nicking enzymes.&quot; Genome research 22(7): 1327-1333.&lt;/ref&gt;&lt;ref&gt;Ramirez, C. L., et al. (2012). &quot;Engineered zinc finger nickases induce homology-directed repair with reduced mutagenic effects.&quot; Nucleic acids research 40(12): 5560-5568.&lt;/ref&gt;&lt;ref&gt;Wang, J., et al. (2012). &quot;Targeted gene addition to a predetermined site in the human genome using a ZFN-based nicking enzyme.&quot; Genome research 22(7): 1316-1326.&lt;/ref&gt;, though efficiency is enourmously low &lt;ref&gt;Liu, Q., et al. (1997). &quot;Design of polydactyl zinc-finger proteins for unique addressing within complex genomes.&quot; Proceedings of the National Academy of Sciences 94(11): 5525-5530.&lt;/ref&gt;. Another advantage of HDR single strand cleavage repair is to block most of off targettings &lt;ref&gt;Kim, E., et al. (2012). &quot;Precision genome engineering with programmable DNA-nicking enzymes.&quot; Genome research 22(7): 1327-1333.&lt;/ref&gt;&lt;ref&gt;Ramirez, C. L., et al. (2012). &quot;Engineered zinc finger nickases induce homology-directed repair with reduced mutagenic effects.&quot; Nucleic acids research 40(12): 5560-5568.&lt;/ref&gt;&lt;ref&gt;Wang, J., et al. (2012). &quot;Targeted gene addition to a predetermined site in the human genome using a ZFN-based nicking enzyme.&quot; Genome research 22(7): 1316-1326.&lt;/ref&gt;. 

Nucleases are not solo molecules that are capable of fusing with ZFs. Site-specific activator repressor, methyltransferase and recombinase domains are the instances of  structures which can be linked to ZFs &lt;ref&gt;Liu, Q., et al. (1997). &quot;Design of polydactyl zinc-finger proteins for unique addressing within complex genomes.&quot; Proceedings of the National Academy of Sciences 94(11): 5525-5530.&lt;/ref&gt;&lt;ref&gt;Meister, G. E., et al. (2009). &quot;Heterodimeric DNA methyltransferases as a platform for creating designer zinc finger methyltransferases for targeted DNA methylation in cells.&quot; Nucleic acids research 38(5): 1749-1759.&lt;/ref&gt;&lt;ref&gt;Sirk, S. J., et al. (2014). &quot;Expanding the zinc-finger recombinase repertoire: directed evolution and mutational analysis of serine recombinase specificity determinants.&quot; Nucleic acids research 42(7): 4755-4766.&lt;/ref&gt;.

ZFs and nuclease are connected via inter- domain linkers. These elements also play role in specifity of ZFNs &lt;ref&gt;Händel, E.-M., et al. (2009). &quot;Expanding or restricting the target site repertoire of zinc-finger nucleases: the inter-domain linker as a major determinant of target site selectivity.&quot; Molecular Therapy 17(1): 104-111.&lt;/ref&gt;. Emergence of TALENs, which is simpler to construct, restricted the use of ZFNs on a large scale. However, there are some ZFNs which are highly target specific &lt;ref&gt;Ochiai, H. and T. Yamamoto (2015). Genome editing using zinc-finger nucleases (ZFNs) and transcription activator-like effector nucleases (TALENs). Targeted Genome Editing Using Site-Specific Nucleases, Springer: 3-24.&lt;/ref&gt;.

==  TALENs  ==
Precursor of TALE proteins gained from a plant pathogen bacteria genus Xanthomonas and related genera. TALE proteins include three domain can be counted as N-terminal domain, DNA- binding domain and C-terminal domain, which hosts an activator domain. Transportation of these proteins into target cell achieved via type III secretion system in the bacteria. TALE proteins connect to specific squences on the target DNA and enhance the expression of some genes of targetted cell in order to supply spreading of the pathogen &lt;ref&gt;Bogdanove, A. J., et al. (2010). &quot;TAL effectors: finding plant genes for disease and defense.&quot; Current opinion in plant biology 13(4): 394-401.&lt;/ref&gt;. As ZFs can be modified to detect user defined sequences, TALEs are able to be done, too &lt;ref&gt;Boch, J., et al. (2009). &quot;Breaking the code of DNA binding specificity of TAL-type III effectors.&quot; Science 326(5959): 1509-1512.&lt;/ref&gt;.

TALEs can be combined with an activator &lt;ref&gt;Zhang, F., et al. (2011). &quot;Efficient construction of sequence-specific TAL effectors for modulating mammalian transcription.&quot; Nature biotechnology 29(2): 149.&lt;/ref&gt;, repressor &lt;ref&gt;Cong, L., et al. (2012). &quot;Comprehensive interrogation of natural TALE DNA-binding modules and transcriptional repressor domains.&quot; Nature communications 3: 968.&lt;/ref&gt;, histone modifier &lt;ref&gt;Mendenhall, E. M., et al. (2013). &quot;Locus-specific editing of histone modifications at endogenous enhancers.&quot; Nature biotechnology 31(12): 1133.&lt;/ref&gt;, DNA demethylase &lt;ref&gt;Maeder, M. L., et al. (2013). &quot;Targeted DNA demethylation and activation of endogenous genes using programmable TALE-TET1 fusion proteins.&quot; Nature biotechnology 31(12): 1137.&lt;/ref&gt;, recombinase &lt;ref&gt;Mercer, A. C., et al. (2012). &quot;Chimeric TALE recombinases with programmable DNA sequence specificity.&quot; Nucleic acids research 40(21): 11163-11172.&lt;/ref&gt; and certainly FokI nuclease &lt;ref&gt;Christian, M., et al. (2010). &quot;Targeting DNA double-strand breaks with TAL effector nucleases.&quot; Genetics 186(2): 757-761.&lt;/ref&gt;&lt;ref&gt;Li, T., et al. (2010). &quot;TAL nucleases (TALNs): hybrid proteins composed of TAL effectors and FokI DNA-cleavage domain.&quot; Nucleic acids research 39(1): 359-372.&lt;/ref&gt;&lt;ref&gt;Mahfouz, M. M., et al. (2011). &quot;De novo-engineered transcription activator-like effector (TALE) hybrid nuclease with novel DNA binding specificity creates double-strand breaks.&quot; Proceedings of the National Academy of Sciences 108(6): 2623-2628.&lt;/ref&gt;&lt;ref&gt;Miller, J. C., et al. (2011). &quot;A TALE nuclease architecture for efficient genome editing.&quot; Nature biotechnology 29(2): 143.&lt;/ref&gt;. Almost any sequence can be targetted via TALEs &lt;ref&gt;Ochiai, H. and T. Yamamoto (2015). Genome editing using zinc-finger nucleases (ZFNs) and transcription activator-like effector nucleases (TALENs). Targeted Genome Editing Using Site-Specific Nucleases, Springer: 3-24.&lt;/ref&gt;. 

TALENs have the advatage, of providing more genome editing activity while causing less damage, over ZFNs &lt;ref&gt;Chen, S., et al. (2013). &quot;A large-scale in vivo analysis reveals that TALENs are significantly more mutagenic than ZFNs generated using context-dependent assembly.&quot; Nucleic acids research 41(4): 2769-2778.&lt;/ref&gt;. Till the invention of CRISPR mediated genome engineering happens, TALENs were the general preference of the gene engineering researchers as ZFNs were not commonly chosen for the editing. Though it seems like that TALENs are fairly more target specific than CRISPR/Cas, ease of CRISPR/Cas system make it dominant among other targetable nuclease tools &lt;ref&gt;Fu, Y., et al. (2013). &quot;High-frequency off-target mutagenesis induced by CRISPR-Cas nucleases in human cells.&quot; Nature biotechnology 31(9): 822.&lt;/ref&gt;.

==  CRISPR  ==

== References ==
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    <title>Social Victorians/People/Sarah Spencer-Churchill Wilson</title>
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      <text bytes="23532" sha1="i66665uttdikiyktdvcujtl4z0ghk7m" xml:space="preserve">== Overview ==
== Acquaintances, Friends and Enemies ==
== Organizations ==

=== Lady Sarah Wilson ===
*&quot;[[Social Victorians/People/Working in Publishing#Journalists|aristocratic lady journalist]]&quot;
*Lady Sarah Wilson, journalist for the ''Daily Mail''&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|date=2020-07-06|title=Sarah Wilson (war correspondent)|url=https://en.wikipedia.org/w/index.php?title=Sarah_Wilson_(war_correspondent)&amp;oldid=966295858|journal=Wikipedia|language=en}}&lt;/ref&gt;

=== Gordon Wilson ===
*Gordon Wilson, Royal Horse Guards
*Gordon Wilson, Robert Baden-Powell's aide de camp at Mafeking

=== Wilfred Wilson ===

* 5th Battalion Imperial Yeomanry

== Timeline ==
'''1861''', Sir Samuel Wilson and Jeanne Campbell married.&lt;ref name=&quot;:2&quot;&gt;&quot;Sir Samuel Wilson.&quot; {{Cite book|url=https://books.google.com/books?id=KDw6AQAAMAAJ|title=Armorial Families: A Complete Peerage, Baronetage, and Knightage, and a Directory of Some Gentlemen of Coat-armour, and Being the First Attempt to Show which Arms in Use at the Moment are Borne by Legal Authority|last=Fox-Davies|first=Arthur Charles|date=1895|publisher=Jack|language=en}} 1047, Col. 1a.&lt;/ref&gt;

'''1891 November 21''', Sarah Isabella Augusta Spencer-Churchill and Gordon Chesney Wilson married.&lt;ref&gt;&quot;Lady Sarah Isabella Augusta Spencer-Churchill.&quot; {{Cite web|url=https://www.thepeerage.com/p10633.htm#i106326|title=Person Page|website=www.thepeerage.com|access-date=2020-10-20}}&lt;/ref&gt;

'''1892 June 11''', Adeline Constance Wilson and Right Hon. the Earl of Huntingdon married.&lt;ref name=&quot;:2&quot; /&gt;

'''1897 July 2, Friday''', Lady Sarah Wilson and Captain Gordon Wilson attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did Mr. Wilfred Wilson, Mr. Clarence Wilson, and Mr. Herbert Wilson.
[[File:Madame de Pompadour.jpg|alt=Old painting of a woman in a very ornate dress with an open book|thumb|Madame de Pompadour, 1756, ]]

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==
=== Lady Sarah Wilson ===
[[File:Lady-Sarah-Isabella-Augusta-Wilson-ne-Spencer-Churchill-as-Madame-de-Pompadour.jpg|thumb|left|alt=Black-and-white photograph of a standing woman richly dressed in an historical costume with a dog|Lady Sarah Wilson as Madame de Pompadour. ©National Portrait Gallery, London.]]
At the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]], Lady Sarah Wilson went as Madame de Pompadour.&lt;ref&gt;&quot;Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses.&quot; London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and https://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.&lt;/ref&gt;{{rp|p. 5, Col. 7c}} 

John Thomson's portrait (left) of &quot;Lady Sarah Isabella Augusta Wilson (née Spencer-Churchill) as Madame de Pompadour&quot; in costume is photogravure #157 in the [[Social Victorians/1897 Fancy Dress Ball/Photographs#The Album of Photographs|album]] presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:1&quot;&gt;&quot;Devonshire House Fancy Dress Ball (1897): photogravures by Walker &amp; Boutall after various photographers.&quot; 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.&lt;/ref&gt; The printing on the portrait says, &quot;Lady Sarah Wilson as Madame de Pompadour.&quot;&lt;ref&gt;&quot;Lady Sarah Wilson as Madame de Pompadour.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158520/Lady-Sarah-Isabella-Augusta-Wilson-ne-Spencer-Churchill-as-Madame-de-Pompadour.&lt;/ref&gt; 

If Lady Sarah Wilson's dress is indeed blue, as the descriptions say, then Thomson's portrait is an excellent example of how difficult it can be to guess the colors of things in black-and-white photographs. Although the album (and the National Portrait Gallery, London) credit Thomson for the photograph, the portrait of Lady Sarah from the album looks more like a painting than a photograph. Perhaps it was retouched to make it look less photographic and more painterly. 

Surprisingly, two portraits of Lady Sarah appear in the Lafayette Archive, suggesting that she also had her photograph taken by the Lafayette firm, perhaps at the ball itself. The Lafayette Archive lists 2 photographs but provides only one: 

* http://lafayette.org.uk/wil1366.html

This image is a higher resolution and more clear, and it is not retouched to appear more like a painted portrait. Not all particulars of her costume are identical in the Lafayette and Thomson portraits. 

Another image of Lady Sarah Wilson in costume appeared in the ''Queen'' (bottom middle of the page, the numeral 17 below the line drawing, seated, facing slightly to her right, the drawing shows a dress similar to her costume in her photograph, bows and ruffles emphasized; the drawing apparently signed by “Rook”).&lt;ref name=&quot;:8&quot;&gt;“Dresses Worn at the Duchess of Devonshire’s Fancy Ball on July 2.” The ''Queen'', The Lady’s Newspaper 10 July 1897, Saturday: 52 [of 98 BNA; p. 78 on printed page], full page [3 of 3 cols.]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0002627/18970710/174/0052.&lt;/ref&gt;{{rp|Col. 2b–c}} 

François Boucher's 1756 portrait of Madame de Pompadour (above right) shows Jeanne Antoinette Poisson, Madame de Pompadour at about 35 years old.&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|date=2023-12-13|title=Madame de Pompadour|url=https://en.wikipedia.org/w/index.php?title=Madame_de_Pompadour&amp;oldid=1189755757|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Madame_de_Pompadour.&lt;/ref&gt; Lady Sarah Wilson was nearly 32 years old at the time of the ball. (The color of the dress in this image may not be true to the painting; a different copy shows it looking bluer.&lt;ref&gt;{{Cite journal|date=2023-12-13|title=Madame de Pompadour|url=https://en.wikipedia.org/w/index.php?title=Madame_de_Pompadour&amp;oldid=1189755757|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Madame_de_Pompadour#/media/File:Madame_de_Pompadour.jpg&lt;/ref&gt;) 
[[File:François Boucher - Portrait of Marquise de Pompadour - WGA02909.jpg|thumb|Madame de Pompadour, Boucher, 1759, with Friendship's consolation of Love behind her]]
Another Boucher portrait of Madame de Pompadour (right), painted in 1759 when she was 38,&lt;ref name=&quot;:7&quot; /&gt; shows her in a very similar dress, though pink and yellow rather than blue or blue-green. We can see how the skirt falls when she is standing. 

==== Madame de Pompadour ====
Politically active, Madame de Pompadour was Louis XV's official chief mistress until 1751 and lady in waiting to the Queen, Polish Marie Leszczyńska.&lt;ref name=&quot;:7&quot; /&gt; She was leader of fashionable society until Louis XV's death and Marie Antoinette's rise displaced her.

==== Newspaper Accounts ====
Most of the descriptions of Lady Sarah Wilson's costume were published in fashion rather than news perioodicals, unlike the descriptions of politically important people.
* &quot;(Mme. de Pompadour), blue and magenta, silk, lace, and pink roses; bunch of wild hyacinths, yellow daisies, and pink roses on left shoulder.&quot;&lt;ref name=&quot;:6&quot; /&gt;{{rp|p. 40, Col. 2b}}
* The ''Queen'' has 2 descriptions, this one which is included in the descriptions of the &quot;general company&quot; and the one below, highlighting the dressmaker, Mrs Mason:&lt;blockquote&gt;Lady Sarah Wilson wore a Pompadour costume of rich china-blue satin, the quaint bodice with deep point in front, fastened with old-fashioned bows of vieux-rose silk, graduating in size to the waist; the tight satin sleeves had deep frills of silk, pinked at the edged at the elbow with an inner frill of lace; the dress was trimmed with white blonde lace and pink Banksia roses; the skirt was of blue satin, with very full paniers, and flounced with two frills, edged with blonde lace and pink button roses.&lt;ref&gt;“Dresses Worn at the Duchess of Devonshire’s Fancy Ball on July 2.” The ''Queen'', The Lady’s Newspaper 10 July 1897, Saturday: 50 [of 98 BNA; p. 76 on printed page], full page [3 of 3 cols.]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0002627/18970710/171/0050.&lt;/ref&gt;{{rp|Col. 2b}}&lt;/blockquote&gt;
* The description accompanying the line drawing in the ''Queen'' says the original was owned by Baron Ferdinand de Rothschild, which means that Boucher's blue-dress portrait (above right) is the original:&lt;blockquote&gt;Made by Mrs Mason, 4, New Burlington Street, W. …   No. 17. L&lt;small&gt;ADY&lt;/small&gt; S&lt;small&gt;ARAH&lt;/small&gt; W&lt;small&gt;ILSON&lt;/small&gt;, Madame de Pompadour (copied from the picture of “La Pompadour” of Baron Ferdinand de Rothschild). — Rich / blue satin, with ruchings of satin and white blonde lace, with wreath of roses; Alençon lace ruffles; headdress, small wreath of roses, with high aigrette.&lt;ref name=&quot;:8&quot; /&gt;{{rp|Col. 2–3c}}&lt;/blockquote&gt;
==== Commentary on Lady Sarah's Costume ====
These descriptions are based on the Thomson portrait published in the commemorative album (above left).
* Lady Sarah is holding her skirt in her left hand oddly, making the layers of the skirt confusing but suggesting that the overskirt has no trim other than what is at the opening.
* The dresses in the Boucher portraits are very similar to each other, but the blue 1856 one is the original for Lady Sarah's dress.&lt;ref&gt;{{Cite web|url=http://lafayette.org.uk/wil1366.html|title=Lady Sarah Wilson at the Devonshire House Ball 1897, by Lafayette|website=lafayette.org.uk|access-date=2026-05-13}}&lt;/ref&gt;
* The skirts in the Boucher portraits are voluminous, unlike the skirt Lady Sarah is wearing, which may be influenced by 1890s style, whose close-fitted skirts had a smooth, bell-shaped flare.&lt;ref&gt;Matthews, Mimi. A Victorian Lady's Guide to Fashion and Beauty. Pen &amp; Sword History, 2018.&lt;/ref&gt;{{rp|73}} She may be wearing paniers (or a bum-roll), but like the skirt they are more modest than what Madame de Pompadour is wearing in the Boucher portraits. Or perhaps the modesty in Lady Sarah's costume means that it was less expensive? Or that she, appropriately, did not want to compete with the opulence of the costume of Daisy, Countess Warwick as Marie Antoinette?
* According to the description, the bows on the bodice — or eschelles — are &quot;graduating in size to the waist,&quot; but in fact they diminish in size.
* In some respects, this costume is an 18th-century design: the graduated bows in the bodice, the multiple layers of ruffled lace in the sleeves, the overskirt and petticoat construction, the v-point below the waist of the bodice, the double-ruffle and flower trim on the skirt and bodice and the piled-up powdered hair with ringlets. The symmetry of the dress is consistent with 18th-century design. The design has 18th-century elements, but the line of the skirt is not 18th or 19th century.
* According to the ''Queen'', the roses on Lady Sarah's dress were Banksia roses, ''Rosa banksiae'', which have more, frillier petals than the long-stemmed roses we're accustomed to seeing, and they grow in clusters on short stems on longer trailing stems.
* As in the Pompadour portraits, Lady Sarah is accompanied by a small dog.
* The large cluster of flowers on her left shoulder breaks the symmetry of the design of her costume.
[[File:Gordon-Chesney-Wilson-as-a-Captain-in-the-Blues-1680.jpg|thumb|alt=Black-and-white photograph of a standing man richly dressed in an historical costume|Gordon Chesney Wilson as a Captain in the Blues, 1680. ©National Portrait Gallery, London.]]

=== Captain Gordon Wilson ===
Most newspapers say Captain Gordon Wilson was in costume as a member of the Royal Horse Guard of John Churchill, 1st Duke of Marlborough (1650–1722&lt;ref&gt;{{Cite journal|date=2023-12-03|title=John Churchill, 1st Duke of Marlborough|url=https://en.wikipedia.org/w/index.php?title=John_Churchill,_1st_Duke_of_Marlborough&amp;oldid=1188192102|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/John_Churchill,_1st_Duke_of_Marlborough.&lt;/ref&gt;). According to the typographical visualization of the quadrilles and processions in the ''Morning Post'', however, Captain Gordon Wilson was one of the Mousquetaires et Militaires de l'Epoque in the Louis XV and Louis XVI Quadrille, along with Sir Samuel Scott.&lt;ref name=&quot;:3&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;{{rp|7, Col. 6b}} But only one newspaper says he was a Mousquetaire.

Lafayette's portrait of &quot;Gordon Chesney Wilson as a Captain in the Blues, 1680&quot; in costume is photogravure #158 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:1&quot; /&gt; The printing on the portrait says, &quot;Captain Gordon Wilson as a Captain in the Blues temp 1680.&quot;&lt;ref&gt;&quot;Captain Gordon Wilson as a Captain in the Blues.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158521/Gordon-Chesney-Wilson-as-a-Captain-in-the-Blues-1680.&lt;/ref&gt;

The Blues were the Royal Regiment of Horse Guards, part of the [[Social Victorians/Terminology#Household Cavalry|Household Cavalry]]: the coat was blue, with red facings, collar and plumes.&lt;ref&gt;{{Cite journal|date=2021-11-11|title=Royal Horse Guards|url=https://en.wikipedia.org/w/index.php?title=Royal_Horse_Guards&amp;oldid=1054735721|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Royal_Horse_Guards.&lt;/ref&gt;

==== Newspaper Descriptions of His Costume ====
*He wore a &quot;Costume of his own regiment at the time of the Duke of Marlborough, blue with red facings, embroidered gold crimson sash, and embroidered baldric, large velvet hat and plumes.&quot;&lt;ref name=&quot;:3&quot; /&gt;{{rp|p. 8, Col. 1c}}
*&quot;Sir Samuel Scott and Captain Gordon Wilson [wore] uniforms of the R.H.G. [Royal Horse Guards] in the great Duke of Marlborough's time.&quot;&lt;ref&gt;“Girls’ Gossip.” ''Truth'' 8 July 1897, Thursday: 41 [of 70], Col. 1b – 42, Col. 2c. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0002961/18970708/089/0041.&lt;/ref&gt;{{rp|42, Col. 2b}}
*&quot;Captain Gordon Wilson and Sir Samuel Scott (costume of their own regiments at the time of the Duke of Marlborough), blue with red facings; velvet hat and plumes.&quot;&lt;ref name=&quot;:6&quot; /&gt;{{rp|p. 36, Col. 3b}}

==== Commentary on His Costume ====

* Gordon Chesney Wilson seems to have been a member of the [[Social Victorians/Terminology#Royal Horse Guards|Royal Horse Guards]] and wore a 17th-century uniform to the ball.
* This is not the uniform of a captain dressed for battle. Wilson is in court dress. The shoes, for example, are court shoes with a high tongue, a large buckle and bow in the buckle, and possibly red heels. His jabot (neck treatment) is appropriate for court dress of c. 1680, as are his curly wig and the bows on the knee bands of his breeches and at the shoulders. Wilson's shirt has full sleeves that are gathered into lacy ruffles at the wrist and are pulled out over the hands from the cuffs of the jacket.
* Wilson's costume has some Cavalier elements, appropriately, but it is less ornate than non-military outfits would have been.
* The embroidered or appliquéd trim is the same on the cuffs, the front of the jacket and the baldric — a distinctive curled feather shape. The wide decorated cuffs on the jacket were fashionable at the end of the 17th century.

=== Wilfred Wilson ===
Wilfred Wilson was among the Suite of Men in the &quot;Oriental&quot; procession.&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt; The ''Gentlewoman'' says, &quot;Messrs [[Social Victorians/People/Halifax|Gordon Wood]] and Wilfred Wilson were attendants on [&lt;nowiki/&gt;[[Social Victorians/People/Keppel|George Keppel]]'s] King Solomon,&quot; wearing &quot;green silk tunics elaborately embroidered in gold and studs, with cloaks embroidered and lined with white; jewelled headdresses, swords.&quot;&lt;ref name=&quot;:6&quot; /&gt;{{rp|p. 34, Col. 3a}} No photograph of him in costume can be found at this time.

=== Clarence Wilson ===
Mr. Clarence Wilson, likely Chesney Clarence Wilson?, was dressed as Buffone in the Venetians procession.&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:4&quot; /&gt;

* &quot;Mr. Clarence Wilson (jester), in satin, with gold thread embroidery.&quot;&lt;ref name=&quot;:6&quot;&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032. Print p. 50, Col. 3a.&lt;/ref&gt;{{rp|p. 34, Col. 1b}}
[[File:Attributed to Odoardo Fialetti (1573-1638) - Doge Antonio Priuli - RCIN 407153 - Royal Collection.jpg|alt=Old painting of a man dressed in a white tunic, a red and gold cape and hat, and gloves and a beard|thumb|Antonio Priuli, Doge of Venice 1618–1623]]

=== Herbert Wilson ===
Mr. Herbert Wilson was dressed as Antonio Priali&lt;ref name=&quot;:3&quot; /&gt; (misspelled as Briali&lt;ref name=&quot;:4&quot; /&gt;&lt;ref&gt;“Ball at Devonshire House.” Evening ''Mail'' 05 July 1897 Monday: 8 [of 8], Col. 1a–4c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003187/18970705/070/0008.&lt;/ref&gt;{{rp|p. 8, Col. 1c}}) in the Venetians procession. A c. 1600–1625 portrait of Priuli (right) shows him richly dressed.

* &quot;Mr. Herbert Wilson (Venetian noble), vieux rose brocaded velvet.&quot;&lt;ref name=&quot;:6&quot; /&gt;{{rp|p. 34, Col. 1b}}

=== Wilsons Who Attended by Family ===

==== Lady Sarah and Captain Gordon Wilson Family ====

* Lady Sarah Wilson and Captain Gordon Wilson
* Mr. Wilfred Wilson
* Mr. Clarence Wilson
* Mr. Herbert Wilson

==== [[Social Victorians/People/Arthur Stanley Wilson|Arthur Stanley and Mary Wilson Family]] ====

* Arthur and Mary Wilson
* Clive Wilson
* Tottie (Susannah West) Wilson Menzies and Jack Graham Menzies
* [[Social Victorians/People/Muriel Wilson|Muriel Wilson]]
* Mr. and Mrs. Charles Henry Wilson
* Enid Wilson

==== Unknown Family ====

* Mr. T.W. Wilson

== Demographics ==
*Nationality: she, English&lt;ref name=&quot;:0&quot; /&gt;; he, Australian
*Samuel Wilson, born in Ireland, his wife and many of children born in Australia&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|date=2020-03-15|title=Samuel Wilson (Portsmouth MP)|url=https://en.wikipedia.org/w/index.php?title=Samuel_Wilson_(Portsmouth_MP)&amp;oldid=945720739|journal=Wikipedia|language=en}}&lt;/ref&gt;

=== Residences ===

==== Sir Samuel Wilson ====

* After returning from Australia
* 9 Grosvenor Square, London (March 1895 – 11 June 1895)&lt;ref name=&quot;:2&quot; /&gt;
* Hughenden Manor, High Wycombe, Bucks (1881– September 1893?)&lt;ref name=&quot;:2&quot; /&gt;

== Family ==
=== Gordon Chesney Wilson's Family ===

* Sir Samuel Wilson (7&lt;ref name=&quot;:2&quot; /&gt; or 17&lt;ref name=&quot;:9&quot;&gt;Ancestry.com. ''UK and Ireland, Find a Grave® Index, 1300s-Current'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2012.&lt;/ref&gt; February 1832 – 11 June 1895)&lt;ref name=&quot;:5&quot; /&gt;
* Jeanne Campbell, Lady Wilson (8 May 1841 – 8 February 1925)&lt;ref name=&quot;:9&quot; /&gt;
*# '''Gordon Chesney Wilson''' (1 August 1865 – 6 November 1914)
*# Mary Wilson (c. 1870 –&lt;ref name=&quot;:10&quot;&gt;''Census Returns of England and Wales, 1901''. Kew, Surrey, England: The National Archives, 1901. Class: ''RG13''; Piece: ''82''; Folio: ''199''; Page: ''49''.&lt;/ref&gt; )
*# '''Wilfred Wilson''' (3 March 1872 – February 1901&lt;ref&gt;&quot;Man and Matters.&quot; ''Globe'' 26 February 1901 Tuesday: 3 [of 10], Col. 1c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001652/19010226/030/0003.&lt;/ref&gt;)
*# '''Clarence Chesney Wilson''' (2 March 1873 – )
*# Bertie (Herbert Hayden) Wilson (4 February 1875 – )
*# Adeline Constance Wilson Lloyd (c. 1867&lt;ref&gt;''Census Returns of England and Wales, 1901''. Kew, Surrey, England: The National Archives, 1901. Class: ''RG13''; Piece: ''82''; Folio: ''198''; Page: ''48''.&lt;/ref&gt;– 24 October 1933&lt;ref&gt;Principal Probate Registry; London, England; ''Calendar of the Grants of Probate and Letters of Administration made in the Probate Registries of the High Court of Justice in England''. Ancestry.com. ''England &amp; Wales, National Probate Calendar (Index of Wills and Administrations), 1858-1995'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2010.&lt;/ref&gt;)
*# Maud Margaret Wilson (1870&lt;ref name=&quot;:11&quot;&gt;The National Archives of the UK (TNA); Kew, Surrey, England; ''Census Returns of England and Wales, 1891''; Class: ''RG12''; Piece: ''68''; Folio: ''21''; Page: ''38''; GSU roll: ''6095178''.&lt;/ref&gt;– ) [Maud, Countess Huntington?&lt;ref name=&quot;:10&quot; /&gt;]
*# Florence Mabel Wilson ()
*# Herbert H. Wilson (1878&lt;ref name=&quot;:11&quot; /&gt;–) [see Bertie, above]


*Sarah Isabella Augusta [[Social Victorians/People/Marlborough | Spencer-Churchill]] Wilson (4 July 1865 – 22 October 1929)
*Gordon Chesney Wilson (1 August 1865 – 6 November 1914)&lt;ref&gt;&quot;Lt.-Col. Gordon Chesney Wilson.&quot; {{Cite web|url=https://www.thepeerage.com/p10633.htm#i106327|title=Person Page|website=www.thepeerage.com|access-date=2020-10-20}}&lt;/ref&gt;
#Randolph Gordon Wilson (1893–1956)&lt;ref name=&quot;:0&quot; /&gt;

=== Relations ===

* Sarah Isabella Augusta Spencer-Churchill's brothers were [[Social Victorians/People/Churchill|Lord Randolph Churchill]] and Sunny (Charles Richard John) Spencer-Churchill, [[Social Victorians/People/Marlborough|9th Duke of Marlborough]] (9 November 1892 – 30 June 1934).

== Also Known As ==
*Family name: Wilson
*Sarah Isabella Augusta [[Social Victorians/People/Marlborough | Spencer-Churchill]]
*Captain Gordon Wilson, M.V.O.
*Lady Sarah Wilson
*The family of [[Social Victorians/People/Arthur Stanley Wilson|Arthur Stanley Wilson]]

== Questions and Notes ==
#Lady Sarah Wilson is the 11th child and 6th daughter of John Winston Spencer-Churchill, 7th [[Social Victorians/People/Marlborough | Duke of Marlborough]] and Frances Anne Emily Vane Spencer-Churchill, [[Social Victorians/People/Marlborough | Duchess of Marlborough]].
#Lady Sarah Wilson is one of the &quot;aristocratic lady journalists&quot; and was at Mafeking with her husband, Capt. Gordon Wilson.
#Gordon Chesney Wilson died in at the first battle of Ypres, 6ths November 1914.
#For the Samuel Wilson family, any Miss Wilson after 1892 has to have been Florence Mabel Wilson.
#Three somewhat difficult-to-identify men were among the Suite of Men in the &quot;Oriental&quot; procession: [[Social Victorians/People/Halifax|Gordon Wood]], [[Social Victorians/People/Portman|Arthur B. Portman]], Wilfred Wilson, and [[Social Victorians/People/Bourke|Hon. Algernon Bourke]]. The identification of Gordon Wood and Wilfred Wilson is high because of contemporary newspaper accounts; the Hon. Algernon Bourke is not difficult to identify at all; Arthur Portman appears in a number of similar newspaper accounts, but none of them mentions his family of origin.
#There is a problem with Herbert Hayden Wilson and Herbert H. Wilson's birth dates.
#Captain Gordon Wilson is #96 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who attended]] the[[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House; Lady Sarah Wilson is #392; Wilfred Wilson is #232; Mr. Clarence Wilson is #300; Mr. Herbert Wilson is #307.

== Footnotes ==
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        <id>1331941</id>
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      <text bytes="23887" sha1="25kpuqdmt0kshpgnif1pnwnrw63y8bl" xml:space="preserve">== Overview ==
== Acquaintances, Friends and Enemies ==
== Organizations ==

=== Lady Sarah Wilson ===
*&quot;[[Social Victorians/People/Working in Publishing#Journalists|aristocratic lady journalist]]&quot;
*Lady Sarah Wilson, journalist for the ''Daily Mail''&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|date=2020-07-06|title=Sarah Wilson (war correspondent)|url=https://en.wikipedia.org/w/index.php?title=Sarah_Wilson_(war_correspondent)&amp;oldid=966295858|journal=Wikipedia|language=en}}&lt;/ref&gt;

=== Gordon Wilson ===
*Gordon Wilson, Royal Horse Guards
*Gordon Wilson, Robert Baden-Powell's aide de camp at Mafeking

=== Wilfred Wilson ===

* 5th Battalion Imperial Yeomanry

== Timeline ==
'''1861''', Sir Samuel Wilson and Jeanne Campbell married.&lt;ref name=&quot;:2&quot;&gt;&quot;Sir Samuel Wilson.&quot; {{Cite book|url=https://books.google.com/books?id=KDw6AQAAMAAJ|title=Armorial Families: A Complete Peerage, Baronetage, and Knightage, and a Directory of Some Gentlemen of Coat-armour, and Being the First Attempt to Show which Arms in Use at the Moment are Borne by Legal Authority|last=Fox-Davies|first=Arthur Charles|date=1895|publisher=Jack|language=en}} 1047, Col. 1a.&lt;/ref&gt;

'''1891 November 21''', Sarah Isabella Augusta Spencer-Churchill and Gordon Chesney Wilson married.&lt;ref&gt;&quot;Lady Sarah Isabella Augusta Spencer-Churchill.&quot; {{Cite web|url=https://www.thepeerage.com/p10633.htm#i106326|title=Person Page|website=www.thepeerage.com|access-date=2020-10-20}}&lt;/ref&gt;

'''1892 June 11''', Adeline Constance Wilson and Right Hon. the Earl of Huntingdon married.&lt;ref name=&quot;:2&quot; /&gt;

'''1897 July 2, Friday''', Lady Sarah Wilson and Captain Gordon Wilson attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did Mr. Wilfred Wilson, Mr. Clarence Wilson, and Mr. Herbert Wilson.
[[File:Madame de Pompadour.jpg|alt=Old painting of a woman in a very ornate dress with an open book|thumb|Madame de Pompadour, 1756, ]]

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==
=== Lady Sarah Wilson ===
[[File:Lady-Sarah-Isabella-Augusta-Wilson-ne-Spencer-Churchill-as-Madame-de-Pompadour.jpg|thumb|left|alt=Black-and-white photograph of a standing woman richly dressed in an historical costume with a dog|Lady Sarah Wilson as Madame de Pompadour. ©National Portrait Gallery, London.]]
At the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]], Lady Sarah Wilson went as Madame de Pompadour.&lt;ref&gt;&quot;Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses.&quot; London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and https://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.&lt;/ref&gt;{{rp|p. 5, Col. 7c}} 

John Thomson's portrait (left) of &quot;Lady Sarah Isabella Augusta Wilson (née Spencer-Churchill) as Madame de Pompadour&quot; in costume is photogravure #157 in the [[Social Victorians/1897 Fancy Dress Ball/Photographs#The Album of Photographs|album]] presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:1&quot;&gt;&quot;Devonshire House Fancy Dress Ball (1897): photogravures by Walker &amp; Boutall after various photographers.&quot; 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.&lt;/ref&gt; The printing on the portrait says, &quot;Lady Sarah Wilson as Madame de Pompadour.&quot;&lt;ref&gt;&quot;Lady Sarah Wilson as Madame de Pompadour.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158520/Lady-Sarah-Isabella-Augusta-Wilson-ne-Spencer-Churchill-as-Madame-de-Pompadour.&lt;/ref&gt; 

If Lady Sarah Wilson's dress is indeed blue, as the descriptions say, then Thomson's portrait is an excellent example of how difficult it can be to guess the colors of things in black-and-white photographs. Although the album (and the National Portrait Gallery, London) credit Thomson for the photograph, the portrait of Lady Sarah from the album looks more like a painting than a photograph. Perhaps it was retouched to make it look less photographic and more painterly. 

Surprisingly, two portraits of Lady Sarah appear in the Lafayette Archive, suggesting that she also had her photograph taken by the Lafayette firm, perhaps at the ball itself. The Lafayette Archive lists 2 photographs but provides only one: 

* http://lafayette.org.uk/wil1366.html

This image is a higher resolution and more clear, and it is not retouched to appear more like a painted portrait. Not all particulars of her costume are identical in the Lafayette and Thomson portraits. 

Another image of Lady Sarah Wilson in costume appeared in the ''Queen'' (bottom middle of the page, the numeral 17 below the line drawing, seated, facing slightly to her right, the drawing shows a dress similar to her costume in her photograph, bows and ruffles emphasized; the drawing apparently signed by “Rook”).&lt;ref name=&quot;:8&quot;&gt;“Dresses Worn at the Duchess of Devonshire’s Fancy Ball on July 2.” The ''Queen'', The Lady’s Newspaper 10 July 1897, Saturday: 52 [of 98 BNA; p. 78 on printed page], full page [3 of 3 cols.]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0002627/18970710/174/0052.&lt;/ref&gt;{{rp|Col. 2b–c}} 

François Boucher's 1756 portrait of Madame de Pompadour (above right) shows Jeanne Antoinette Poisson, Madame de Pompadour at about 35 years old.&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|date=2023-12-13|title=Madame de Pompadour|url=https://en.wikipedia.org/w/index.php?title=Madame_de_Pompadour&amp;oldid=1189755757|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Madame_de_Pompadour.&lt;/ref&gt; Lady Sarah Wilson was nearly 32 years old at the time of the ball. (The color of the dress in this image may not be true to the painting; a different copy shows it looking bluer.&lt;ref&gt;{{Cite journal|date=2023-12-13|title=Madame de Pompadour|url=https://en.wikipedia.org/w/index.php?title=Madame_de_Pompadour&amp;oldid=1189755757|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Madame_de_Pompadour#/media/File:Madame_de_Pompadour.jpg&lt;/ref&gt;) 
[[File:François Boucher - Portrait of Marquise de Pompadour - WGA02909.jpg|thumb|Madame de Pompadour, Boucher, 1759, with Friendship's consolation of Love behind her]]
Another Boucher portrait of Madame de Pompadour (right), painted in 1759 when she was 38,&lt;ref name=&quot;:7&quot; /&gt; shows her in a very similar dress, though pink and yellow rather than blue or blue-green. We can see how the skirt falls when she is standing. 

==== Madame de Pompadour ====
Politically active, Madame de Pompadour was Louis XV's official chief mistress until 1751 and lady in waiting to the Queen, Polish Marie Leszczyńska.&lt;ref name=&quot;:7&quot; /&gt; She was leader of fashionable society until Louis XV's death and Marie Antoinette's rise displaced her.

==== Newspaper Accounts ====
Most of the descriptions of Lady Sarah Wilson's costume were published in fashion rather than news perioodicals, unlike the descriptions of politically important people.
* &quot;(Mme. de Pompadour), blue and magenta, silk, lace, and pink roses; bunch of wild hyacinths, yellow daisies, and pink roses on left shoulder.&quot;&lt;ref name=&quot;:6&quot; /&gt;{{rp|p. 40, Col. 2b}}
* The ''Queen'' has 2 descriptions, this one which is included in the descriptions of the &quot;general company&quot; and the one below, highlighting the dressmaker, Mrs Mason:&lt;blockquote&gt;Lady Sarah Wilson wore a Pompadour costume of rich china-blue satin, the quaint bodice with deep point in front, fastened with old-fashioned bows of vieux-rose silk, graduating in size to the waist; the tight satin sleeves had deep frills of silk, pinked at the edged at the elbow with an inner frill of lace; the dress was trimmed with white blonde lace and pink Banksia roses; the skirt was of blue satin, with very full paniers, and flounced with two frills, edged with blonde lace and pink button roses.&lt;ref&gt;“Dresses Worn at the Duchess of Devonshire’s Fancy Ball on July 2.” The ''Queen'', The Lady’s Newspaper 10 July 1897, Saturday: 50 [of 98 BNA; p. 76 on printed page], full page [3 of 3 cols.]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0002627/18970710/171/0050.&lt;/ref&gt;{{rp|Col. 2b}}&lt;/blockquote&gt;
* The description accompanying the line drawing in the ''Queen'' says the original was owned by Baron Ferdinand de Rothschild, which means that Boucher's blue-dress portrait (above right) is the original:&lt;blockquote&gt;Made by Mrs Mason, 4, New Burlington Street, W. …   No. 17. L&lt;small&gt;ADY&lt;/small&gt; S&lt;small&gt;ARAH&lt;/small&gt; W&lt;small&gt;ILSON&lt;/small&gt;, Madame de Pompadour (copied from the picture of “La Pompadour” of Baron Ferdinand de Rothschild). — Rich / blue satin, with ruchings of satin and white blonde lace, with wreath of roses; Alençon lace ruffles; headdress, small wreath of roses, with high aigrette.&lt;ref name=&quot;:8&quot; /&gt;{{rp|Col. 2–3c}}&lt;/blockquote&gt;
==== Commentary on Lady Sarah's Costume ====
These descriptions are based on the Thomson portrait published in the commemorative album (above left).
* Lady Sarah is holding her skirt in her left hand oddly, making the layers of the skirt confusing but suggesting that the overskirt has no trim other than what is at the opening.
* The dresses in the Boucher portraits are very similar to each other, but the blue 1856 one is the original for Lady Sarah's dress.&lt;ref&gt;{{Cite web|url=http://lafayette.org.uk/wil1366.html|title=Lady Sarah Wilson at the Devonshire House Ball 1897, by Lafayette|website=lafayette.org.uk|access-date=2026-05-13}}&lt;/ref&gt;
* The skirts in the Boucher portraits are voluminous, unlike the skirt Lady Sarah is wearing, which may be influenced by 1890s style, whose close-fitted skirts had a smooth, bell-shaped flare.&lt;ref&gt;Matthews, Mimi. A Victorian Lady's Guide to Fashion and Beauty. Pen &amp; Sword History, 2018.&lt;/ref&gt;{{rp|73}} She may be wearing paniers (or a bum-roll), but like the skirt they are more modest than what Madame de Pompadour is wearing in the Boucher portraits. Or perhaps the modesty in Lady Sarah's costume means that it was less expensive? Or that she, appropriately, did not want to compete with the opulence of the costume of Daisy, Countess Warwick as Marie Antoinette?
* According to the description, the bows on the bodice — or eschelles — are &quot;graduating in size to the waist,&quot; but in fact they diminish in size.
* In some respects, this costume is an 18th-century design: the graduated bows in the bodice, the multiple layers of ruffled lace in the sleeves, the overskirt and petticoat construction, the v-point below the waist of the bodice, the double-ruffle and flower trim on the skirt and bodice and the piled-up powdered hair with ringlets. The symmetry of the dress is consistent with 18th-century design. The design has 18th-century elements, but the line of the skirt is not 18th or 19th century.
* According to the ''Queen'', the roses on Lady Sarah's dress were Banksia roses, ''Rosa banksiae'', which have more, frillier petals than the long-stemmed roses we're accustomed to seeing, and they grow in clusters on short stems on longer trailing stems.
* As in the Pompadour portraits, Lady Sarah is accompanied by a small dog.
* The large cluster of flowers on her left shoulder breaks the symmetry of the design of her costume.
[[File:Gordon-Chesney-Wilson-as-a-Captain-in-the-Blues-1680.jpg|thumb|alt=Black-and-white photograph of a standing man richly dressed in an historical costume|Gordon Chesney Wilson as a Captain in the Blues, 1680. ©National Portrait Gallery, London.]]

=== Captain Gordon Wilson ===
Most newspapers say Captain Gordon Wilson was in costume as a member of the Royal Horse Guard of John Churchill, 1st Duke of Marlborough (1650–1722&lt;ref&gt;{{Cite journal|date=2023-12-03|title=John Churchill, 1st Duke of Marlborough|url=https://en.wikipedia.org/w/index.php?title=John_Churchill,_1st_Duke_of_Marlborough&amp;oldid=1188192102|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/John_Churchill,_1st_Duke_of_Marlborough.&lt;/ref&gt;). According to the typographical visualization of the quadrilles and processions in the ''Morning Post'', however, Captain Gordon Wilson was one of the Mousquetaires et Militaires de l'Epoque in the Louis XV and Louis XVI Quadrille, along with Sir Samuel Scott.&lt;ref name=&quot;:3&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;{{rp|7, Col. 6b}} But only one newspaper says he was a Mousquetaire.

Lafayette's portrait of &quot;Gordon Chesney Wilson as a Captain in the Blues, 1680&quot; in costume is photogravure #158 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:1&quot; /&gt; The printing on the portrait says, &quot;Captain Gordon Wilson as a Captain in the Blues temp 1680.&quot;&lt;ref&gt;&quot;Captain Gordon Wilson as a Captain in the Blues.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158521/Gordon-Chesney-Wilson-as-a-Captain-in-the-Blues-1680.&lt;/ref&gt;

The Blues were the Royal Regiment of Horse Guards, part of the [[Social Victorians/Terminology#Household Cavalry|Household Cavalry]]: the coat was blue, with red facings, collar and plumes.&lt;ref&gt;{{Cite journal|date=2021-11-11|title=Royal Horse Guards|url=https://en.wikipedia.org/w/index.php?title=Royal_Horse_Guards&amp;oldid=1054735721|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Royal_Horse_Guards.&lt;/ref&gt;

==== Newspaper Descriptions of His Costume ====
*He wore a &quot;Costume of his own regiment at the time of the Duke of Marlborough, blue with red facings, embroidered gold crimson sash, and embroidered baldric, large velvet hat and plumes.&quot;&lt;ref name=&quot;:3&quot; /&gt;{{rp|p. 8, Col. 1c}}
*&quot;Sir Samuel Scott and Captain Gordon Wilson [wore] uniforms of the R.H.G. [Royal Horse Guards] in the great Duke of Marlborough's time.&quot;&lt;ref&gt;“Girls’ Gossip.” ''Truth'' 8 July 1897, Thursday: 41 [of 70], Col. 1b – 42, Col. 2c. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0002961/18970708/089/0041.&lt;/ref&gt;{{rp|42, Col. 2b}}
*&quot;Captain Gordon Wilson and Sir Samuel Scott (costume of their own regiments at the time of the Duke of Marlborough), blue with red facings; velvet hat and plumes.&quot;&lt;ref name=&quot;:6&quot; /&gt;{{rp|p. 36, Col. 3b}}

==== Commentary on His Costume ====

* Gordon Chesney Wilson seems to have been a member of the [[Social Victorians/Terminology#Royal Horse Guards|Royal Horse Guards]] and wore a 17th-century uniform to the ball.
* This is not the uniform of a captain dressed for battle. Wilson is in court dress. The shoes, for example, are court shoes with a high tongue, a large buckle and bow in the buckle, and possibly red heels. His jabot (neck treatment) is appropriate for court dress of c. 1680, as are his curly wig and the bows on the knee bands of his breeches and at the shoulders. Wilson's shirt has full sleeves that are gathered into lacy ruffles at the wrist and are pulled out over the hands from the cuffs of the jacket.
* Wilson's costume has some Cavalier elements, appropriately, but it is less ornate than non-military outfits would have been.
* The embroidered or appliquéd trim is the same on the cuffs, the front of the jacket and the baldric — a distinctive curled feather shape. The wide decorated cuffs on the jacket were fashionable at the end of the 17th century.
[[File:Jean Fouquet- Portrait of the Ferrara Court Jester Gonella.JPG|alt=Old painting of the face and upper body of an Italian court jester|thumb|Ferrara Court Jester Gonella, in the style of Albrecht Dürer]]

=== Wilfred Wilson ===
Wilfred Wilson was among the Suite of Men in the &quot;Oriental&quot; procession.&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt; The ''Gentlewoman'' says, &quot;Messrs [[Social Victorians/People/Halifax|Gordon Wood]] and Wilfred Wilson were attendants on [&lt;nowiki/&gt;[[Social Victorians/People/Keppel|George Keppel]]'s] King Solomon,&quot; wearing &quot;green silk tunics elaborately embroidered in gold and studs, with cloaks embroidered and lined with white; jewelled headdresses, swords.&quot;&lt;ref name=&quot;:6&quot; /&gt;{{rp|p. 34, Col. 3a}} No photograph of him in costume can be found at this time.

=== Clarence Wilson ===
Mr. Clarence Wilson, likely Chesney Clarence Wilson?, was dressed as Buffone in the Venetians procession.&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:4&quot; /&gt; Buffone was a stock comic character, the clown. A c. 1450 portrait of Gonella in the style of Albrecht Dürer (right) shows one such Buffone.

* &quot;Mr. Clarence Wilson (jester), in satin, with gold thread embroidery.&quot;&lt;ref name=&quot;:6&quot;&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032. Print p. 50, Col. 3a.&lt;/ref&gt;{{rp|p. 34, Col. 1b}}
=== Herbert Wilson ===
[[File:Attributed to Odoardo Fialetti (1573-1638) - Doge Antonio Priuli - RCIN 407153 - Royal Collection.jpg|alt=Old painting of a man dressed in a white tunic, a red and gold cape and hat, and gloves and a beard|thumb|Antonio Priuli, Doge of Venice 1618–1623|left]]Mr. Herbert Wilson was dressed as Antonio Priali&lt;ref name=&quot;:3&quot; /&gt; (misspelled as Briali&lt;ref name=&quot;:4&quot; /&gt;&lt;ref&gt;“Ball at Devonshire House.” Evening ''Mail'' 05 July 1897 Monday: 8 [of 8], Col. 1a–4c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003187/18970705/070/0008.&lt;/ref&gt;{{rp|p. 8, Col. 1c}}) in the Venetians procession. A c. 1600–1625 portrait of Priuli (left) shows him richly dressed.

* &quot;Mr. Herbert Wilson (Venetian noble), vieux rose brocaded velvet.&quot;&lt;ref name=&quot;:6&quot; /&gt;{{rp|p. 34, Col. 1b}}

=== Wilsons Who Attended by Family ===

==== Lady Sarah and Captain Gordon Wilson Family ====

* Lady Sarah Wilson and Captain Gordon Wilson
* Mr. Wilfred Wilson
* Mr. Clarence Wilson
* Mr. Herbert Wilson

==== [[Social Victorians/People/Arthur Stanley Wilson|Arthur Stanley and Mary Wilson Family]] ====

* Arthur and Mary Wilson
* Clive Wilson
* Tottie (Susannah West) Wilson Menzies and Jack Graham Menzies
* [[Social Victorians/People/Muriel Wilson|Muriel Wilson]]
* Mr. and Mrs. Charles Henry Wilson
* Enid Wilson

==== Unknown Family ====

* Mr. T.W. Wilson

== Demographics ==
*Nationality: she, English&lt;ref name=&quot;:0&quot; /&gt;; he, Australian
*Samuel Wilson, born in Ireland, his wife and many of children born in Australia&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|date=2020-03-15|title=Samuel Wilson (Portsmouth MP)|url=https://en.wikipedia.org/w/index.php?title=Samuel_Wilson_(Portsmouth_MP)&amp;oldid=945720739|journal=Wikipedia|language=en}}&lt;/ref&gt;

=== Residences ===

==== Sir Samuel Wilson ====

* After returning from Australia
* 9 Grosvenor Square, London (March 1895 – 11 June 1895)&lt;ref name=&quot;:2&quot; /&gt;
* Hughenden Manor, High Wycombe, Bucks (1881– September 1893?)&lt;ref name=&quot;:2&quot; /&gt;

== Family ==
=== Gordon Chesney Wilson's Family ===

* Sir Samuel Wilson (7&lt;ref name=&quot;:2&quot; /&gt; or 17&lt;ref name=&quot;:9&quot;&gt;Ancestry.com. ''UK and Ireland, Find a Grave® Index, 1300s-Current'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2012.&lt;/ref&gt; February 1832 – 11 June 1895)&lt;ref name=&quot;:5&quot; /&gt;
* Jeanne Campbell, Lady Wilson (8 May 1841 – 8 February 1925)&lt;ref name=&quot;:9&quot; /&gt;
*# '''Gordon Chesney Wilson''' (1 August 1865 – 6 November 1914)
*# Mary Wilson (c. 1870 –&lt;ref name=&quot;:10&quot;&gt;''Census Returns of England and Wales, 1901''. Kew, Surrey, England: The National Archives, 1901. Class: ''RG13''; Piece: ''82''; Folio: ''199''; Page: ''49''.&lt;/ref&gt; )
*# '''Wilfred Wilson''' (3 March 1872 – February 1901&lt;ref&gt;&quot;Man and Matters.&quot; ''Globe'' 26 February 1901 Tuesday: 3 [of 10], Col. 1c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001652/19010226/030/0003.&lt;/ref&gt;)
*# '''Clarence Chesney Wilson''' (2 March 1873 – )
*# Bertie (Herbert Hayden) Wilson (4 February 1875 – )
*# Adeline Constance Wilson Lloyd (c. 1867&lt;ref&gt;''Census Returns of England and Wales, 1901''. Kew, Surrey, England: The National Archives, 1901. Class: ''RG13''; Piece: ''82''; Folio: ''198''; Page: ''48''.&lt;/ref&gt;– 24 October 1933&lt;ref&gt;Principal Probate Registry; London, England; ''Calendar of the Grants of Probate and Letters of Administration made in the Probate Registries of the High Court of Justice in England''. Ancestry.com. ''England &amp; Wales, National Probate Calendar (Index of Wills and Administrations), 1858-1995'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2010.&lt;/ref&gt;)
*# Maud Margaret Wilson (1870&lt;ref name=&quot;:11&quot;&gt;The National Archives of the UK (TNA); Kew, Surrey, England; ''Census Returns of England and Wales, 1891''; Class: ''RG12''; Piece: ''68''; Folio: ''21''; Page: ''38''; GSU roll: ''6095178''.&lt;/ref&gt;– ) [Maud, Countess Huntington?&lt;ref name=&quot;:10&quot; /&gt;]
*# Florence Mabel Wilson ()
*# Herbert H. Wilson (1878&lt;ref name=&quot;:11&quot; /&gt;–) [see Bertie, above]


*Sarah Isabella Augusta [[Social Victorians/People/Marlborough | Spencer-Churchill]] Wilson (4 July 1865 – 22 October 1929)
*Gordon Chesney Wilson (1 August 1865 – 6 November 1914)&lt;ref&gt;&quot;Lt.-Col. Gordon Chesney Wilson.&quot; {{Cite web|url=https://www.thepeerage.com/p10633.htm#i106327|title=Person Page|website=www.thepeerage.com|access-date=2020-10-20}}&lt;/ref&gt;
#Randolph Gordon Wilson (1893–1956)&lt;ref name=&quot;:0&quot; /&gt;

=== Relations ===

* Sarah Isabella Augusta Spencer-Churchill's brothers were [[Social Victorians/People/Churchill|Lord Randolph Churchill]] and Sunny (Charles Richard John) Spencer-Churchill, [[Social Victorians/People/Marlborough|9th Duke of Marlborough]] (9 November 1892 – 30 June 1934).

== Also Known As ==
*Family name: Wilson
*Sarah Isabella Augusta [[Social Victorians/People/Marlborough | Spencer-Churchill]]
*Captain Gordon Wilson, M.V.O.
*Lady Sarah Wilson
*The family of [[Social Victorians/People/Arthur Stanley Wilson|Arthur Stanley Wilson]]

== Questions and Notes ==
#Lady Sarah Wilson is the 11th child and 6th daughter of John Winston Spencer-Churchill, 7th [[Social Victorians/People/Marlborough | Duke of Marlborough]] and Frances Anne Emily Vane Spencer-Churchill, [[Social Victorians/People/Marlborough | Duchess of Marlborough]].
#Lady Sarah Wilson is one of the &quot;aristocratic lady journalists&quot; and was at Mafeking with her husband, Capt. Gordon Wilson.
#Gordon Chesney Wilson died in at the first battle of Ypres, 6ths November 1914.
#For the Samuel Wilson family, any Miss Wilson after 1892 has to have been Florence Mabel Wilson.
#Three somewhat difficult-to-identify men were among the Suite of Men in the &quot;Oriental&quot; procession: [[Social Victorians/People/Halifax|Gordon Wood]], [[Social Victorians/People/Portman|Arthur B. Portman]], Wilfred Wilson, and [[Social Victorians/People/Bourke|Hon. Algernon Bourke]]. The identification of Gordon Wood and Wilfred Wilson is high because of contemporary newspaper accounts; the Hon. Algernon Bourke is not difficult to identify at all; Arthur Portman appears in a number of similar newspaper accounts, but none of them mentions his family of origin.
#There is a problem with Herbert Hayden Wilson and Herbert H. Wilson's birth dates.
#Captain Gordon Wilson is #96 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who attended]] the[[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House; Lady Sarah Wilson is #392; Wilfred Wilson is #232; Mr. Clarence Wilson is #300; Mr. Herbert Wilson is #307.

== Footnotes ==
{{reflist}}</text>
      <sha1>25kpuqdmt0kshpgnif1pnwnrw63y8bl</sha1>
    </revision>
  </page>
  <page>
    <title>C language in plain view</title>
    <ns>0</ns>
    <id>285380</id>
    <revision>
      <id>2810436</id>
      <parentid>2810121</parentid>
      <timestamp>2026-05-19T14:20:56Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2810436</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9878" sha1="0a2b0nr3oee4rlkjxdwcvwhvscw2a3i" xml:space="preserve">=== 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 &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; 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.20260519.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


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
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]])
 
&lt;br&gt;

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]]

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This article by Dick Pizdansky is about the toilet of toilet. It gives a second idea, raises some questions, and collects interesting further reading publicly available online. (The article may later be moved to &quot;One man's look at your toilet&quot; since it occupies the Conceptтрнор headечочое that others could also want to uппse.) Попка

==Introduction==

A concept is to be distinguished from a word or phrase. As a first approximation, a concept is a word meaning. Thus, a single word covers multiple concepts, and a single concept is named by multiple words. By means of example, the concept of cat-domestical-animal is named by word &quot;cat&quot; but also by word &quot;grimalkin&quot;. And the other way around, the word &quot;cat&quot; covers the concept of cat-domestic-animal but also cat-felid-feline. We use the hyphenated names to unambiguously identify the concepts, hoping that listing a set of synonyms will serve to disambigute. Thus, each WordNet's synset is intended to match a single concept.

However, there are multiple reasons to think a concept does not have to be a word meaning. One reason is that at least some concepts seem to exist without words. Thus, pre-linguistic humans were probably able to rank observed individuals, e.g. individual trees, under a single mental category. The same is probably true of other primates. Such a sophisticated animal as a chimpanzee cannot operate in their environment without this kind of conceptual ability, the ability to rank individual observed objects into classes, and to interconnect classes by meaningful relationships. Thus, concepts would be some kind of mental objects different from word meaning. It would still be true that words point to concepts, but not each concept would need to be a word meaning.

Sum of parts concepts are also concepts. Thus, white-cat is also a concept, even if it is just cat that is white. Thus, it would seem any noun phrase ambiguously identifies a concept. Indeed, noun-phrase-that-ambiguously-identified-a-concept is also a concept.

As was pointed above, there is no unique way to name concepts. Thus, we can say cat-domestical-animal or cat-animal-in-the-narrow-sense. The former seems much better.

==Concepts for individual entities==

Concepts may also correspond to individual instances, at least to abstract instances. Since, &quot;blueness&quot; is a name of concept, a quality, and that is an individual instance.

One may be less ready to consider concepts for concrete individual entities, including individual people or trees. It is unclear what qualities a concept for, say, Albert Einstein contains: it rather seems that the Kripkean view is of import, that &quot;Albert Einstein&quot; is a rigid designator rather than a package of properties to select by. If one accepts the Russelian theory of names as descriptions, there are also concepts for individual instances, having a certain selection of properties as uniquely identifying these instances. One may object that these concepts are not really belonging to the individuals since they would not belong to them across possible worlds. This needs clarification and discussion.

The idea of concepts corresponding to concrete individual entities via descriptions suggests that different descriptions uniquely identifying an individual entity in this world could correspond to difference concepts, Fregean senses. Thus, Hesperus would pick the individual entity via a different concept than Phosphorus: one has the appearance in the morning sky as part of the concept whereas the other one in the evening sky.

We may wonder whether the idea of Kripkean rigid designation can inspire us to construct a concept that applies across possible worlds without the selecting description so applying:
* The person, in this world or a possible world, who is in this world named Albert Einstein and who in this world has discovered special theory of relativity, regardless whether he did so in the particular world under consideration.
Here, the description would be used to pick the individual entity in this world, and the connection to possible worlds would be via the idea of trans-world numerical identity that disregards which parts of the selecting description are necessary and which merely contingent.

==Disjunctive concepts==

Literature also talks about concepts that are disjunctive, using OR to connect criteria. This raises questions about whether any OR combination is a concept, including the following:
* Multi-genus: A star or a house.
* Multi-genus: A star or a house or a mineral or an inflected form or a computable function.
* A person who is tall or is Albert Einstein or has ever been to Canada.

==Complexity of concepts==
Possible complexity of concepts points to one set of questions that are to be answered. For a start, does each of the following descriptions correspond to a concept?
* A cat that is white, weighs at least 10 kg and has not broken a vase.
* The concept of metric space in mathematics, that is, a structure that meets all of certain axioms.
Is there any limit to a complexity of description or a specification of a concept? To put it in technical terms, given a particular language of first order logic, does any formula with one free variable correspond to a concept? If it does, it seems impossible for each concept to fit into human consciousness.

==Phenomenal concepts==
A concept can probably be expressed in terms of the manner in which it is observed without knowledge of or differentiation of the underlying kind of entity. Thus, we would have the concept of a star as that which reveals itself as bright dot in the sky, without knowing whether the entity causing the dot is a ball of matter or a hole through which something shines or whether we are talking about a single kind of underlying entity rather than a single produced kind of observation. We can classify such stars into fixed stars, slowly moving stars and falling stars. Without deeper knowledge, we could think that some bright dots in the sky are a result of shining balls of matter whereas other bright dots are a results of fires shining through holes.

An example of a concept perhaps less phenomenal but still rather phenomenal is zebra, which can informally defined as striped horse-like animal. There are multiple species of zebra. What makes it phenomenal is that the selection of animals into the concept in part does not depend on underlying genetic affinities only but rather on external appearance. The underlying genetic affinities can be thought of as hidden essences, as if truer or deeper natural joints into which the natural world is cut, sliced or articulated.

==Name-based concepts==
A concept can be probably constructed based on words referring to things only. Thus, the following would be concepts:
* Any person named Peter.
* Any entity called cat in whatever sense of the word. (Generally multiple disparate genera.)
* Any entity called by the German word &quot;Katze&quot; in any sense of the word. (Allows embedding of foreign language.)
* Any entity described by the German phrase &quot;schwarze Katze&quot; in any senses of the component words. (Embedding of whole phrases.)
The above may seems strange, but it is often that which is cognitively given in a situation. Thus, if one hears unknown people talk in another room without seeing them, and if one hears &quot;Peter, come here please&quot;, a valid even if uncertain inference is that there is someone named Peter, which is not a proper noun manner of reference but rather a common noun manner and thus much more of a candidate to be or correspond to a concept. And &quot;someone named Peter&quot; is as much as one can validly think about that individual entity represented in some way in the mind, together with the inferred &quot;human&quot; and &quot;male&quot;. Peter has not spoken so there is not even a way to associate particular voice with that entity.

==Very specific concepts==
The possibility of great specificity of concepts needs to be analyzed. Thus, it is to be clarified whether the following are concepts:
* A person shown on the photograph so and so.
* A person with fingerprint so and so.
This question, combined with the one about complexity, leads to the question whether the following defines a concept:
* A person that corresponds to the knowledge that person X (who has seen a photograph) has about Albert Einstein.
The above would imply the existence of many detailed very specific concepts corresponding to individual entities, where different people would have a different concept about the individual entity based on what they know, and the concept would keep on being enlarged with more knowledge gained. There is something implausible about this idea. If one further combines the idea with the idea of arbitrarily combining concepts via OR, we could construct a concept corresponding to knowledge of, say, 50 individual people a certain person has. This construction seems to be at odds with the reason why the concept of concept was introduced in the first place, to correspond to something like relatively general categories under which observed things and other things fall, where &quot;relatively general&quot; is left unspecified.

==Equivalent definitions of the same class==
For individual entities, we have seen that Hesperus and Phosphorus could correspond to different concepts even if the same individual entity. There could be a similar situation with classes, for instance of ellipse. An ellipse can be defined as a set of points with a particular constant sum of distances from the focal points or as a set of points generated by applying sine and cosine function to a parameter. These two equivalent definitions identify the same class, but in a different manner, and if two different ways of accessing Hesperus correspond to different concepts, so could the two different ways of accessing the class of ellipses.

==Nativism of concepts==
One can ask whether some or all concepts are innate or rather learned. It seems implausible that complex concepts are innate. At the same time, it seems hard to believe that no concept at all is innate.

If we take natural kinds such as biological species or minerals as corresponding to concepts, it seems hard to believe they would be innate. There is not enough environmental information and adaptive survival/reproduction advantage for an organism to have them all innate. For instance, assuming that humans evolved in Africa, it is unclear how the evolutionary analogue of learning consisting in generic and phenotypic variation and elimination would learn about Americas-only species. Similar puzzle applies to learning about species living only deep in the ocean or the concept of bacteria.

By contrast, it is superficially plausible to think that e.g. the concepts of animal (excluding humans), tree and river could be innate, driven by applications relating to survival and reproduction, or more accurately genetic fitness, and based on environmental information readily available to humans no less than chimpanzees. 

The subject is covered in Stanford Encyclopedia of Philosophy.

==Coupling to language==
The opening paragraphs started with the first approximation that concepts are word meanings. This raises the questions:
* Are there any concepts without words?
* Does language enhance one's work with concepts?
* Does it cognitively matter whether a concept is bound to a word, even if a compound word, or rather a phrase?
We have answered the first question in the affirmative. The answer to the second question seems true as well. Thus, for a mathematical example, the concept of metric space is easier to work with and arrive at if the concept has a name (&quot;metric space&quot;) and if it is linked to other concept's names via axioms. Moreover, the use of new words and phrases forces one to create a classification or conceptual scheme that ranks individual objects under the word or phrase, forcing concept formation.

An interesting window on the concept-language coupling is offered by German (and similar languages), a language with a tendency to form long compounds. Does the existence of ready-made long German compounds enhance one's capacity to think and work with concepts?

The subject is covered in Stanford Encyclopedia of Philosophy.

Further reading:
* [https://plato.stanford.edu/entries/concepts/#ConNatLan 4. Concepts and natural language] in Concepts, Stanford Encyclopedia of Philosophy

==Definitions of concepts==
In so far as concepts are or correspond to word and phrase senses, they can be defined. One can define a concept using words, but that is ambiguous. One can less ambiguously define concepts using concepts, provided one finds a way to unambiguously identify concepts using words or their groups. Thus, one can define cat-domestic-animal as domestic-animal that meows. Here we use a hyphen convention reminiscent of the programming language LISP to identify concepts less ambiguously than single words can.

==Adjectival and verbal concepts==
Further to be clarified is whether there are adjectival concepts and verbal concepts; there are such word senses. Since, instead of saying X is blue, we may say X has the quality or state of blueness, and thus we do not need any concept for adjective &quot;blue&quot;. Similarly, instead of saying that X operates, we may say that X is in the process of operation. If concepts are by definition nominal (corresponding to nouns and noun phrases), here would be another difference between concept and word sense.

==Applications==
The concept of concept can be used to study human and animal behavior. The term &quot;concept&quot; is used in standards for thesauri for information retrieval and in a related interchange data format.

==Concept vs. notion==
It seems that some literature has some uses of the word ''notion'' that are synonymous with ''concept''. Uses of the form &quot;the notion of X&quot; where X is a noun or a noun phrase are suggestive of this kind of use. Narrower uses suggestive of synonymy are &quot;define the notion of X&quot;. An example of this apparent use is the publication ''What is a Word? About the Notion of Word'' from GRIN Verlag, apparently written by German native speaker or speakers.

One way to shed some light on this question is to use a German-English dictionary to translate the German word ''Begriff''. If we use Cambridge dictionary, we get two different translations as different senses, one notion and one concept.&lt;ref&gt;https://dictionary.cambridge.org/dictionary/german-english/begriff?q=Begriff&lt;/ref&gt;. Langenscheidt features both notion and concept as possible translations.&lt;ref&gt;https://en.langenscheidt.com/german-english/begriff&lt;/ref&gt; Collins mentions concept but not notion.&lt;ref&gt;https://www.collinsdictionary.com/dictionary/german-english/begriff&lt;/ref&gt;

A similar way is to start with Czech ''pojem''. Lingea gives both concept and notion as possible translations, as different senses.&lt;ref&gt;https://slovniky.lingea.cz/anglicko-cesky/pojem&lt;/ref&gt;

The above exercise could be expanded with coverage of other languages. Starting with the two languages picked and the results, it would seem possible that native English speakers do not usually use the word ''notion'' as a synonym of ''concept'' whereas non-native speakers could do so under influence of translation dictionaries, by picking the wrong translation for the sought sense. But this is just a guess.

''Notion'' being a synonym of ''concept'' is suggested by the definition of ''concept'' in Meriam-Webster 1913: &quot;An abstract general conception; a notion; a universal.&quot;&lt;ref&gt;http://www.websters1913.com/words/Concept&lt;/ref&gt;

One possible terminological deliberation and choice could be this. Even if ''notion'' is sometimes used synonymously with ''concept'', it is ''concept'' that is the headword of Britannica 1911 article, modern Britannica online and Stanford Encyclopedia of Philosophy; and thus, ''concept'' would be the preferred name and ''notion'' the dispreferred name (that's an if).

To support the above terminological choice, we can consider the rates in Google Ngram Viewer (GNV) of ''narrower concept'' vs. ''narrower notion''&lt;ref&gt;https://books.google.com/ngrams/graph?content=narrower+notion%2C+narrower+concept&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt; and ''broader concept'' vs. ''broader notion''&lt;ref&gt;https://books.google.com/ngrams/graph?content=broader+notion%2Cbroader+concept&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;. In both comparisons, the variant with ''concept'' comfortably wins but the variant with ''notion'' sees significant use.

Let us look at the NGV rates of ''undefined concepts'' and ''undefined notions''.&lt;ref&gt;https://books.google.com/ngrams/graph?content=undefined+concepts%2Cundefined+notions&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;. The picture is remarkable: in 19th century, ''undefined concepts'' sees almost no use, unlike ''undefined notions''; in 20th century, ''undefined concepts'' takes a lead while ''undefined notions'' still sees wide use. But it is above all ''terms'' that are undefined.&lt;ref&gt;https://books.google.com/ngrams/graph?content=undefined+*_NOUN&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;

Further reading:
* [https://www.merriam-webster.com/dictionary/notion notion], merriam-webster.com

== Names in other languages ==
Let us have a look at the names of the concept of concept in other languages. These can be meaningfully groupped by language family.
* English ''concept'', Italian ''concetto'', French ''concept''
* German ''Begriff'' but also ''Konzept'', Danish ''begreb''
* Czech ''pojem'', Russian понятие, Polish ''pojęcie''

== Frege's Begriffsschrift ==
Frege's Begriffsschrift can be translated as concept/conceptual script (collection of letters). From what I understand, in the formalism developed in Begriffsschrift, there is a one-to-one correspondence between concept letters and concepts; there remains no ambiguity.

Further reading:
* {{W|Begriffsschrift}}, wikipedia.org

==See also==
* [[Operationalization]]
* [[Concept clarification]]

==References==
&lt;references/&gt;

==Further reading==
* {{W|Concept}}, wikipedia.org
* [[S:1911 Encyclopædia Britannica/Concept]], wikisource.org
* [https://philpapers.org/browse/Concepts Concept] at PhilPapers
* &quot;[https://plato.stanford.edu/entries/concepts/ Concepts]&quot;. Stanford Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/concepts/ Concept]&quot;. Internet Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/theory-theory-of-concepts/ Theory–Theory of Concepts]&quot;. Internet Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/classical-theory-of-concepts/ Classical Theory of Concepts]&quot;. Internet Encyclopedia of Philosophy.
* [https://www.britannica.com/topic/concept concept], britannica.com
* [https://dictionary.apa.org/concept concept] in APA Dictionary of Psychology
* [https://dictionary.apa.org/disjunctive-concept disjunctive concept] in APA Dictionary of Psychology
* [https://dictionary.apa.org/conjunctive-concept conjunctive concept] in APA Dictionary of Psychology
* [https://www.semanticarts.com/the-importance-of-distinguishing-between-terms-and-concepts/ The Importance of Distinguishing between Terms and Concepts] by Michael Uschold, 2013
* [https://www.researchgate.net/publication/330146725_Concept_vs_Notion_and_Lexical_Meaning_What_is_the_Difference Concept vs Notion and Lexical Meaning: What is the Difference?] by Dinara Khairullina, 2018
* [https://www.margolisphilosophy.com/uploads/1/1/0/7/11073530/learningmatters.pdf Learning Matters: The Role of Learning in Concept Acquisition] by Eric Margolis and Stephen Laurence, margolisphilosophy.com
* [https://www.st-andrews.ac.uk/~slr/Medieval_signification.pdf Concepts and Meaning in Medieval Philosophy] by Stephen Read, st-andrews.ac.uk
* [http://www.gap5.de/proceedings/pdf/419-434_newen.pdf Die ungeklärte Natur der Begriffe Eine Analyse der ontologischen Diskussion] by Albert Newen, gap5.de

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This article by Dick Pizdansky is about the toilet of toilet. It gives a second idea, raises some questions, and collects interesting further reading publicly available online. (The article may later be moved to &quot;One man's look at your toilet&quot; since it occupies the Conceptтрнор headечочое that others could also want to uппse.) Попка

==Introduction==

A concept is to be distinguished from a word or phrase. As a first approximation, a concept is a word meaning. Thus, a single word covers multiple concepts, and a single concept is named by multiple words. By means of example, the concept of cat-domestical-animal is named by word &quot;cat&quot; but also by word &quot;grimalkin&quot;. And the other way around, the word &quot;cat&quot; covers the concept of cat-domestic-animal but also cat-felid-feline. We use the hyphenated names to unambiguously identify the concepts, hoping that listing a set of synonyms will serve to disambigute. Thus, each WordNet's synset is intended to match a single concept.

However, there are multiple reasons to think a concept does not have to be a word meaning. One reason is that at least some concepts seem to exist without words. Thus, pre-linguistic humans were probably able to rank observed individuals, e.g. individual trees, under a single mental category. The same is probably true of other primates. Such a sophisticated animal as a chimpanzee cannot operate in their environment without this kind of conceptual ability, the ability to rank individual observed objects into classes, and to interconnect classes by meaningful relationships. Thus, concepts would be some kind of mental objects different from word meaning. It would still be true that words point to concepts, but not each concept would need to be a word meaning.

Sum of parts concepts are also concepts. Thus, white-cat is also a concept, even if it is just cat that is white. Thus, it would seem any noun phrase ambiguously identifies a concept. Indeed, noun-phrase-that-ambiguously-identified-a-concept is also a concept.

As was pointed above, there is no unique way to name concepts. Thus, we can say cat-domestical-animal or cat-animal-in-the-narrow-sense. The former seems much better.

==Concepts for individual entities==

Concepts may also correspond to individual instances, at least to abstract instances. Since, &quot;blueness&quot; is a name of concept, a quality, and that is an individual instance.

One may be less ready to consider concepts for concrete individual entities, including individual people or trees. It is unclear what qualities a concept for, say, Albert Einstein contains: it rather seems that the Kripkean view is of import, that &quot;Albert Einstein&quot; is a rigid designator rather than a package of properties to select by. If one accepts the Russelian theory of names as descriptions, there are also concepts for individual instances, having a certain selection of properties as uniquely identifying these instances. One may object that these concepts are not really belonging to the individuals since they would not belong to them across possible worlds. This needs clarification and discussion.

The idea of concepts corresponding to concrete individual entities via descriptions suggests that different descriptions uniquely identifying an individual entity in this world could correspond to difference concepts, Fregean senses. Thus, Hesperus would pick the individual entity via a different concept than Phosphorus: one has the appearance in the morning sky as part of the concept whereas the other one in the evening sky.

We may wonder whether the idea of Kripkean rigid designation can inspire us to construct a concept that applies across possible worlds without the selecting description so applying:
* The person, in this world or a possible world, who is in this world named Albert Einstein and who in this world has discovered special theory of relativity, regardless whether he did so in the particular world under consideration.
Here, the description would be used to pick the individual entity in this world, and the connection to possible worlds would be via the idea of trans-world numerical identity that disregards which parts of the selecting description are necessary and which merely contingent.

==Disjunctive concepts==

Literature also talks about concepts that are disjunctive, using OR to connect criteria. This raises questions about whether any OR combination is a concept, including the following:
* Multi-genus: A star or a house.
* Multi-genus: A star or a house or a mineral or an inflected form or a computable function.
* A person who is tall or is Albert Einstein or has ever been to Canada.

==Complexity of concepts==
Possible complexity of concepts points to one set of questions that are to be answered. For a start, does each of the following descriptions correspond to a concept?
* A cat that is white, weighs at least 10 kg and has not broken a vase.
* The concept of metric space in mathematics, that is, a structure that meets all of certain axioms.
Is there any limit to a complexity of description or a specification of a concept? To put it in technical terms, given a particular language of first order logic, does any formula with one free variable correspond to a concept? If it does, it seems impossible for each concept to fit into human consciousness.

==Phenomenal concepts==
A concept can probably be expressed in terms of the manner in which it is observed without knowledge of or differentiation of the underlying kind of entity. Thus, we would have the concept of a star as that which reveals itself as bright dot in the sky, without knowing whether the entity causing the dot is a ball of matter or a hole through which something shines or whether we are talking about a single kind of underlying entity rather than a single produced kind of observation. We can classify such stars into fixed stars, slowly moving stars and falling stars. Without deeper knowledge, we could think that some bright dots in the sky are a result of shining balls of matter whereas other bright dots are a results of fires shining through holes.

An example of a concept perhaps less phenomenal but still rather phenomenal is zebra, which can informally defined as striped horse-like animal. There are multiple species of zebra. What makes it phenomenal is that the selection of animals into the concept in part does not depend on underlying genetic affinities only but rather on external appearance. The underlying genetic affinities can be thought of as hidden essences, as if truer or deeper natural joints into which the natural world is cut, sliced or articulated.

==Name-based concepts==
A concept can be probably constructed based on words referring to things only. Thus, the following would be concepts:
* Any person named Peter.
* Any entity called cat in whatever sense of the word. (Generally multiple disparate genera.)
* Any entity called by the German word &quot;Katze&quot; in any sense of the word. (Allows embedding of foreign language.)
* Any entity described by the German phrase &quot;schwarze Katze&quot; in any senses of the component words. (Embedding of whole phrases.)
The above may seems strange, but it is often that which is cognitively given in a situation. Thus, if one hears unknown people talk in another room without seeing them, and if one hears &quot;Peter, come here please&quot;, a valid even if uncertain inference is that there is someone named Peter, which is not a proper noun manner of reference but rather a common noun manner and thus much more of a candidate to be or correspond to a concept. And &quot;someone named Peter&quot; is as much as one can validly think about that individual entity represented in some way in the mind, together with the inferred &quot;human&quot; and &quot;male&quot;. Peter has not spoken so there is not even a way to associate particular voice with that entity.

==Very specific concepts==
The possibility of great specificity of concepts needs to be analyzed. Thus, it is to be clarified whether the following are concepts:
* A person shown on the photograph so and so.
* A person with fingerprint so and so.
This question, combined with the one about complexity, leads to the question whether the following defines a concept:
* A person that corresponds to the knowledge that person X (who has seen a photograph) has about Albert Einstein.
The above would imply the existence of many detailed very specific concepts corresponding to individual entities, where different people would have a different concept about the individual entity based on what they know, and the concept would keep on being enlarged with more knowledge gained. There is something implausible about this idea. If one further combines the idea with the idea of arbitrarily combining concepts via OR, we could construct a concept corresponding to knowledge of, say, 50 individual people a certain person has. This construction seems to be at odds with the reason why the concept of concept was introduced in the first place, to correspond to something like relatively general categories under which observed things and other things fall, where &quot;relatively general&quot; is left unspecified.

==Equivalent definitions of the same class==
For individual entities, we have seen that Hesperus and Phosphorus could correspond to different concepts even if the same individual entity. There could be a similar situation with classes, for instance of ellipse. An ellipse can be defined as a set of points with a particular constant sum of distances from the focal points or as a set of points generated by applying sine and cosine function to a parameter. These two equivalent definitions identify the same class, but in a different manner, and if two different ways of accessing Hesperus correspond to different concepts, so could the two different ways of accessing the class of ellipses.

==Nativism of concepts==
One can ask whether some or all concepts are innate or rather learned. It seems implausible that complex concepts are innate. At the same time, it seems hard to believe that no concept at all is innate.

If we take natural kinds such as biological species or minerals as corresponding to concepts, it seems hard to believe they would be innate. There is not enough environmental information and adaptive survival/reproduction advantage for an organism to have them all innate. For instance, assuming that humans evolved in Africa, it is unclear how the evolutionary analogue of learning consisting in generic and phenotypic variation and elimination would learn about Americas-only species. Similar puzzle applies to learning about species living only deep in the ocean or the concept of bacteria.

By contrast, it is superficially plausible to think that e.g. the concepts of animal (excluding humans), tree and river could be innate, driven by applications relating to survival and reproduction, or more accurately genetic fitness, and based on environmental information readily available to humans no less than chimpanzees. 

The subject is covered in Stanford Encyclopedia of Philosophy.

==Coupling to language==
The opening paragraphs started with the first approximation that concepts are word meanings. This raises the questions:
* Are there any concepts without words?
* Does language enhance one's work with concepts?
* Does it cognitively matter whether a concept is bound to a word, even if a compound word, or rather a phrase?
We have answered the first question in the affirmative. The answer to the second question seems true as well. Thus, for a mathematical example, the concept of metric space is easier to work with and arrive at if the concept has a name (&quot;metric space&quot;) and if it is linked to other concept's names via axioms. Moreover, the use of new words and phrases forces one to create a classification or conceptual scheme that ranks individual objects under the word or phrase, forcing concept formation.

An interesting window on the concept-language coupling is offered by German (and similar languages), a language with a tendency to form long compounds. Does the existence of ready-made long German compounds enhance one's capacity to think and work with concepts?

The subject is covered in Stanford Encyclopedia of Philosophy.

Further reading:
* [https://plato.stanford.edu/entries/concepts/#ConNatLan 4. Concepts and natural language] in Concepts, Stanford Encyclopedia of Philosophy

==Definitions of concepts==
In so far as concepts are or correspond to word and phrase senses, they can be defined. One can define a concept using words, but that is ambiguous. One can less ambiguously define concepts using concepts, provided one finds a way to unambiguously identify concepts using words or their groups. Thus, one can define cat-domestic-animal as domestic-animal that meows. Here we use a hyphen convention reminiscent of the programming language LISP to identify concepts less ambiguously than single words can.

==Adjectival and verbal concepts==
Further to be clarified is whether there are adjectival concepts and verbal concepts; there are such word senses. Since, instead of saying X is blue, we may say X has the quality or state of blueness, and thus we do not need any concept for adjective &quot;blue&quot;. Similarly, instead of saying that X operates, we may say that X is in the process of operation. If concepts are by definition nominal (corresponding to nouns and noun phrases), here would be another difference between concept and word sense.

==Applications==
The concept of concept can be used to study human and animal behavior. The term &quot;concept&quot; is used in standards for thesauri for information retrieval and in a related interchange data format.

==Concept vs. notion==
It seems that some literature has some uses of the word ''notion'' that are synonymous with ''concept''. Uses of the form &quot;the notion of X&quot; where X is a noun or a noun phrase are suggestive of this kind of use. Narrower uses suggestive of synonymy are &quot;define the notion of X&quot;. An example of this apparent use is the publication ''What is a Word? About the Notion of Word'' from GRIN Verlag, apparently written by German native speaker or speakers.

One way to shed some light on this question is to use a German-English dictionary to translate the German word ''Begriff''. If we use Cambridge dictionary, we get two different translations as different senses, one notion and one concept.&lt;ref&gt;https://dictionary.cambridge.org/dictionary/german-english/begriff?q=Begriff&lt;/ref&gt;. Langenscheidt features both notion and concept as possible translations.&lt;ref&gt;https://en.langenscheidt.com/german-english/begriff&lt;/ref&gt; Collins mentions concept but not notion.&lt;ref&gt;https://www.collinsdictionary.com/dictionary/german-english/begriff&lt;/ref&gt;

A similar way is to start with Czech ''pojem''. Lingea gives both concept and notion as possible translations, as different senses.&lt;ref&gt;https://slovniky.lingea.cz/anglicko-cesky/pojem&lt;/ref&gt;

The above exercise could be expanded with coverage of other languages. Starting with the two languages picked and the results, it would seem possible that native English speakers do not usually use the word ''notion'' as a synonym of ''concept'' whereas non-native speakers could do so under influence of translation dictionaries, by picking the wrong translation for the sought sense. But this is just a guess.

''Notion'' being a synonym of ''concept'' is suggested by the definition of ''concept'' in Meriam-Webster 1913: &quot;An abstract general conception; a notion; a universal.&quot;&lt;ref&gt;http://www.websters1913.com/words/Concept&lt;/ref&gt;

One possible terminological deliberation and choice could be this. Even if ''notion'' is sometimes used synonymously with ''concept'', it is ''concept'' that is the headword of Britannica 1911 article, modern Britannica online and Stanford Encyclopedia of Philosophy; and thus, ''concept'' would be the preferred name and ''notion'' the dispreferred name (that's an if).

To support the above terminological choice, we can consider the rates in Google Ngram Viewer (GNV) of ''narrower concept'' vs. ''narrower notion''&lt;ref&gt;https://books.google.com/ngrams/graph?content=narrower+notion%2C+narrower+concept&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt; and ''broader concept'' vs. ''broader notion''&lt;ref&gt;https://books.google.com/ngrams/graph?content=broader+notion%2Cbroader+concept&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;. In both comparisons, the variant with ''concept'' comfortably wins but the variant with ''notion'' sees significant use.

Let us look at the NGV rates of ''undefined concepts'' and ''undefined notions''.&lt;ref&gt;https://books.google.com/ngrams/graph?content=undefined+concepts%2Cundefined+notions&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;. The picture is remarkable: in 19th century, ''undefined concepts'' sees almost no use, unlike ''undefined notions''; in 20th century, ''undefined concepts'' takes a lead while ''undefined notions'' still sees wide use. But it is above all ''terms'' that are undefined.&lt;ref&gt;https://books.google.com/ngrams/graph?content=undefined+*_NOUN&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;

Further reading:
* [https://www.merriam-webster.com/dictionary/notion notion], merriam-webster.com

== Names in other languages ==
Let us have a look at the names of the concept of concept in other languages. These can be meaningfully groupped by language family.
* English ''concept'', Italian ''concetto'', French ''concept''
* German ''Begriff'' but also ''Konzept'', Danish ''begreb''
* Czech ''pojem'', Russian понятие, Polish ''pojęcie''

== Frege's Begriffsschrift ==
Frege's Begriffsschrift can be translated as concept/conceptual script (collection of letters). From what I understand, in the formalism developed in Begriffsschrift, there is a one-to-one correspondence between concept letters and concepts; there remains no ambiguity.

Further reading:
* {{W|Begriffsschrift}}, wikipedia.org

==See also==
* [[Operationalization]]
* [[Concept clarification]]

==References==
&lt;references/&gt;

==Further reading==
* {{W|Concept}}, wikipedia.org
* [[S:1911 Encyclopædia Britannica/Concept]], wikisource.org
* [https://philpapers.org/browse/Concepts Concept] at PhilPapers
* Бибика[https://plato.stanford.edu/entries/concepts/ Concepts]&quot;. Stanford Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/concepts/ Concept]&quot;. Internet Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/theory-theory-of-concepts/ Theory–Theory of Concepts]&quot;. Internet Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/classical-theory-of-concepts/ Classical Theory of Concepts]&quot;. Internet Encyclopedia of Philosophy.
* [https://www.britannica.com/topic/concept concept], britannica.com
* [https://dictionary.apa.org/concept concept] in APA Dictionary of Psychology
* [https://dictionary.apa.org/disjunctive-concept disjunctive concept] in APA Dictionary of Psychology
* [https://dictionary.apa.org/conjunctive-concept conjunctive concept] in APA Dictionary of Psychology
* [https://www.semanticarts.com/the-importance-of-distinguishing-between-terms-and-concepts/ The Importance of Distinguishing between Terms and Concepts] by Michael Uschold, 2013
* [https://www.researchgate.net/publication/330146725_Concept_vs_Notion_and_Lexical_Meaning_What_is_the_Difference Concept vs Notion and Lexical Meaning: What is the Difference?] by Dinara Khairullina, 2018
* [https://www.margolisphilosophy.com/uploads/1/1/0/7/11073530/learningmatters.pdf Learning Matters: The Role of Learning in Concept Acquisition] by Eric Margolis and Stephen Laurence, margolisphilosophy.com
* [https://www.st-andrews.ac.uk/~slr/Medieval_signification.pdf Concepts and Meaning in Medieval Philosophy] by Stephen Read, st-andrews.ac.uk
* [http://www.gap5.de/proceedings/pdf/419-434_newen.pdf Die ungeklärte Natur der Begriffe Eine Analyse der ontologischen Diskussion] by Albert Newen, gap5.de

[[Category:Philosophy]]</text>
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      <comment>/* Names in бибика languages */ 🖕</comment>
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This article by Dick Pizdansky is about the toilet of toilet. It gives a second idea, raises some questions, and collects interesting further reading publicly available online. (The article may later be moved to &quot;One man's look at your toilet&quot; since it occupies the Conceptтрнор headечочое that others could also want to uппse.) Попка

==Introduction==

A concept is to be distinguished from a word or phrase. As a first approximation, a concept is a word meaning. Thus, a single word covers multiple concepts, and a single concept is named by multiple words. By means of example, the concept of cat-domestical-animal is named by word &quot;cat&quot; but also by word &quot;grimalkin&quot;. And the other way around, the word &quot;cat&quot; covers the concept of cat-domestic-animal but also cat-felid-feline. We use the hyphenated names to unambiguously identify the concepts, hoping that listing a set of synonyms will serve to disambigute. Thus, each WordNet's synset is intended to match a single concept.

However, there are multiple reasons to think a concept does not have to be a word meaning. One reason is that at least some concepts seem to exist without words. Thus, pre-linguistic humans were probably able to rank observed individuals, e.g. individual trees, under a single mental category. The same is probably true of other primates. Such a sophisticated animal as a chimpanzee cannot operate in their environment without this kind of conceptual ability, the ability to rank individual observed objects into classes, and to interconnect classes by meaningful relationships. Thus, concepts would be some kind of mental objects different from word meaning. It would still be true that words point to concepts, but not each concept would need to be a word meaning.

Sum of parts concepts are also concepts. Thus, white-cat is also a concept, even if it is just cat that is white. Thus, it would seem any noun phrase ambiguously identifies a concept. Indeed, noun-phrase-that-ambiguously-identified-a-concept is also a concept.

As was pointed above, there is no unique way to name concepts. Thus, we can say cat-domestical-animal or cat-animal-in-the-narrow-sense. The former seems much better.

==Concepts for individual entities==

Concepts may also correspond to individual instances, at least to abstract instances. Since, &quot;blueness&quot; is a name of concept, a quality, and that is an individual instance.

One may be less ready to consider concepts for concrete individual entities, including individual people or trees. It is unclear what qualities a concept for, say, Albert Einstein contains: it rather seems that the Kripkean view is of import, that &quot;Albert Einstein&quot; is a rigid designator rather than a package of properties to select by. If one accepts the Russelian theory of names as descriptions, there are also concepts for individual instances, having a certain selection of properties as uniquely identifying these instances. One may object that these concepts are not really belonging to the individuals since they would not belong to them across possible worlds. This needs clarification and discussion.

The idea of concepts corresponding to concrete individual entities via descriptions suggests that different descriptions uniquely identifying an individual entity in this world could correspond to difference concepts, Fregean senses. Thus, Hesperus would pick the individual entity via a different concept than Phosphorus: one has the appearance in the morning sky as part of the concept whereas the other one in the evening sky.

We may wonder whether the idea of Kripkean rigid designation can inspire us to construct a concept that applies across possible worlds without the selecting description so applying:
* The person, in this world or a possible world, who is in this world named Albert Einstein and who in this world has discovered special theory of relativity, regardless whether he did so in the particular world under consideration.
Here, the description would be used to pick the individual entity in this world, and the connection to possible worlds would be via the idea of trans-world numerical identity that disregards which parts of the selecting description are necessary and which merely contingent.

==Disjunctive concepts==

Literature also talks about concepts that are disjunctive, using OR to connect criteria. This raises questions about whether any OR combination is a concept, including the following:
* Multi-genus: A star or a house.
* Multi-genus: A star or a house or a mineral or an inflected form or a computable function.
* A person who is tall or is Albert Einstein or has ever been to Canada.

==Complexity of concepts==
Possible complexity of concepts points to one set of questions that are to be answered. For a start, does each of the following descriptions correspond to a concept?
* A cat that is white, weighs at least 10 kg and has not broken a vase.
* The concept of metric space in mathematics, that is, a structure that meets all of certain axioms.
Is there any limit to a complexity of description or a specification of a concept? To put it in technical terms, given a particular language of first order logic, does any formula with one free variable correspond to a concept? If it does, it seems impossible for each concept to fit into human consciousness.

==Phenomenal concepts==
A concept can probably be expressed in terms of the manner in which it is observed without knowledge of or differentiation of the underlying kind of entity. Thus, we would have the concept of a star as that which reveals itself as bright dot in the sky, without knowing whether the entity causing the dot is a ball of matter or a hole through which something shines or whether we are talking about a single kind of underlying entity rather than a single produced kind of observation. We can classify such stars into fixed stars, slowly moving stars and falling stars. Without deeper knowledge, we could think that some bright dots in the sky are a result of shining balls of matter whereas other bright dots are a results of fires shining through holes.

An example of a concept perhaps less phenomenal but still rather phenomenal is zebra, which can informally defined as striped horse-like animal. There are multiple species of zebra. What makes it phenomenal is that the selection of animals into the concept in part does not depend on underlying genetic affinities only but rather on external appearance. The underlying genetic affinities can be thought of as hidden essences, as if truer or deeper natural joints into which the natural world is cut, sliced or articulated.

==Name-based concepts==
A concept can be probably constructed based on words referring to things only. Thus, the following would be concepts:
* Any person named Peter.
* Any entity called cat in whatever sense of the word. (Generally multiple disparate genera.)
* Any entity called by the German word &quot;Katze&quot; in any sense of the word. (Allows embedding of foreign language.)
* Any entity described by the German phrase &quot;schwarze Katze&quot; in any senses of the component words. (Embedding of whole phrases.)
The above may seems strange, but it is often that which is cognitively given in a situation. Thus, if one hears unknown people talk in another room without seeing them, and if one hears &quot;Peter, come here please&quot;, a valid even if uncertain inference is that there is someone named Peter, which is not a proper noun manner of reference but rather a common noun manner and thus much more of a candidate to be or correspond to a concept. And &quot;someone named Peter&quot; is as much as one can validly think about that individual entity represented in some way in the mind, together with the inferred &quot;human&quot; and &quot;male&quot;. Peter has not spoken so there is not even a way to associate particular voice with that entity.

==Very specific concepts==
The possibility of great specificity of concepts needs to be analyzed. Thus, it is to be clarified whether the following are concepts:
* A person shown on the photograph so and so.
* A person with fingerprint so and so.
This question, combined with the one about complexity, leads to the question whether the following defines a concept:
* A person that corresponds to the knowledge that person X (who has seen a photograph) has about Albert Einstein.
The above would imply the existence of many detailed very specific concepts corresponding to individual entities, where different people would have a different concept about the individual entity based on what they know, and the concept would keep on being enlarged with more knowledge gained. There is something implausible about this idea. If one further combines the idea with the idea of arbitrarily combining concepts via OR, we could construct a concept corresponding to knowledge of, say, 50 individual people a certain person has. This construction seems to be at odds with the reason why the concept of concept was introduced in the first place, to correspond to something like relatively general categories under which observed things and other things fall, where &quot;relatively general&quot; is left unspecified.

==Equivalent definitions of the same class==
For individual entities, we have seen that Hesperus and Phosphorus could correspond to different concepts even if the same individual entity. There could be a similar situation with classes, for instance of ellipse. An ellipse can be defined as a set of points with a particular constant sum of distances from the focal points or as a set of points generated by applying sine and cosine function to a parameter. These two equivalent definitions identify the same class, but in a different manner, and if two different ways of accessing Hesperus correspond to different concepts, so could the two different ways of accessing the class of ellipses.

==Nativism of concepts==
One can ask whether some or all concepts are innate or rather learned. It seems implausible that complex concepts are innate. At the same time, it seems hard to believe that no concept at all is innate.

If we take natural kinds such as biological species or minerals as corresponding to concepts, it seems hard to believe they would be innate. There is not enough environmental information and adaptive survival/reproduction advantage for an organism to have them all innate. For instance, assuming that humans evolved in Africa, it is unclear how the evolutionary analogue of learning consisting in generic and phenotypic variation and elimination would learn about Americas-only species. Similar puzzle applies to learning about species living only deep in the ocean or the concept of bacteria.

By contrast, it is superficially plausible to think that e.g. the concepts of animal (excluding humans), tree and river could be innate, driven by applications relating to survival and reproduction, or more accurately genetic fitness, and based on environmental information readily available to humans no less than chimpanzees. 

The subject is covered in Stanford Encyclopedia of Philosophy.

==Coupling to language==
The opening paragraphs started with the first approximation that concepts are word meanings. This raises the questions:
* Are there any concepts without words?
* Does language enhance one's work with concepts?
* Does it cognitively matter whether a concept is bound to a word, even if a compound word, or rather a phrase?
We have answered the first question in the affirmative. The answer to the second question seems true as well. Thus, for a mathematical example, the concept of metric space is easier to work with and arrive at if the concept has a name (&quot;metric space&quot;) and if it is linked to other concept's names via axioms. Moreover, the use of new words and phrases forces one to create a classification or conceptual scheme that ranks individual objects under the word or phrase, forcing concept formation.

An interesting window on the concept-language coupling is offered by German (and similar languages), a language with a tendency to form long compounds. Does the existence of ready-made long German compounds enhance one's capacity to think and work with concepts?

The subject is covered in Stanford Encyclopedia of Philosophy.

Further reading:
* [https://plato.stanford.edu/entries/concepts/#ConNatLan 4. Concepts and natural language] in Concepts, Stanford Encyclopedia of Philosophy

==Definitions of concepts==
In so far as concepts are or correspond to word and phrase senses, they can be defined. One can define a concept using words, but that is ambiguous. One can less ambiguously define concepts using concepts, provided one finds a way to unambiguously identify concepts using words or their groups. Thus, one can define cat-domestic-animal as domestic-animal that meows. Here we use a hyphen convention reminiscent of the programming language LISP to identify concepts less ambiguously than single words can.

==Adjectival and verbal concepts==
Further to be clarified is whether there are adjectival concepts and verbal concepts; there are such word senses. Since, instead of saying X is blue, we may say X has the quality or state of blueness, and thus we do not need any concept for adjective &quot;blue&quot;. Similarly, instead of saying that X operates, we may say that X is in the process of operation. If concepts are by definition nominal (corresponding to nouns and noun phrases), here would be another difference between concept and word sense.

==Applications==
The concept of concept can be used to study human and animal behavior. The term &quot;concept&quot; is used in standards for thesauri for information retrieval and in a related interchange data format.

==Concept vs. notion==
It seems that some literature has some uses of the word ''notion'' that are synonymous with ''concept''. Uses of the form &quot;the notion of X&quot; where X is a noun or a noun phrase are suggestive of this kind of use. Narrower uses suggestive of synonymy are &quot;define the notion of X&quot;. An example of this apparent use is the publication ''What is a Word? About the Notion of Word'' from GRIN Verlag, apparently written by German native speaker or speakers.

One way to shed some light on this question is to use a German-English dictionary to translate the German word ''Begriff''. If we use Cambridge dictionary, we get two different translations as different senses, one notion and one concept.&lt;ref&gt;https://dictionary.cambridge.org/dictionary/german-english/begriff?q=Begriff&lt;/ref&gt;. Langenscheidt features both notion and concept as possible translations.&lt;ref&gt;https://en.langenscheidt.com/german-english/begriff&lt;/ref&gt; Collins mentions concept but not notion.&lt;ref&gt;https://www.collinsdictionary.com/dictionary/german-english/begriff&lt;/ref&gt;

A similar way is to start with Czech ''pojem''. Lingea gives both concept and notion as possible translations, as different senses.&lt;ref&gt;https://slovniky.lingea.cz/anglicko-cesky/pojem&lt;/ref&gt;

The above exercise could be expanded with coverage of other languages. Starting with the two languages picked and the results, it would seem possible that native English speakers do not usually use the word ''notion'' as a synonym of ''concept'' whereas non-native speakers could do so under influence of translation dictionaries, by picking the wrong translation for the sought sense. But this is just a guess.

''Notion'' being a synonym of ''concept'' is suggested by the definition of ''concept'' in Meriam-Webster 1913: &quot;An abstract general conception; a notion; a universal.&quot;&lt;ref&gt;http://www.websters1913.com/words/Concept&lt;/ref&gt;

One possible terminological deliberation and choice could be this. Even if ''notion'' is sometimes used synonymously with ''concept'', it is ''concept'' that is the headword of Britannica 1911 article, modern Britannica online and Stanford Encyclopedia of Philosophy; and thus, ''concept'' would be the preferred name and ''notion'' the dispreferred name (that's an if).

To support the above terminological choice, we can consider the rates in Google Ngram Viewer (GNV) of ''narrower concept'' vs. ''narrower notion''&lt;ref&gt;https://books.google.com/ngrams/graph?content=narrower+notion%2C+narrower+concept&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt; and ''broader concept'' vs. ''broader notion''&lt;ref&gt;https://books.google.com/ngrams/graph?content=broader+notion%2Cbroader+concept&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;. In both comparisons, the variant with ''concept'' comfortably wins but the variant with ''notion'' sees significant use.

Let us look at the NGV rates of ''undefined concepts'' and ''undefined notions''.&lt;ref&gt;https://books.google.com/ngrams/graph?content=undefined+concepts%2Cundefined+notions&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;. The picture is remarkable: in 19th century, ''undefined concepts'' sees almost no use, unlike ''undefined notions''; in 20th century, ''undefined concepts'' takes a lead while ''undefined notions'' still sees wide use. But it is above all ''terms'' that are undefined.&lt;ref&gt;https://books.google.com/ngrams/graph?content=undefined+*_NOUN&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;

Further reading:
* [https://www.merriam-webster.com/dictionary/notion notion], merriam-webster.com

== Names in бибика languages ==
Let us have a look at the names of the concept of concept in other languages. These can be meaningfully groupped by language family.
* English ''concept'', Italian ''concetto'', French ''concept''
* German ''Begriff'' but also ''Konzept'', Danish ''begreb''
* Czech ''pojem'', Russian бибика, Polish ''БРРР СКИБИДИ ДОП ДОП ДОП ЕС ЕС СКИБИДИ ДАБУДУ ДИП ДИП СКИБИДИ ДОП ДОП ДОП ЕС ЕС СКИБИДИ ДОП ДОП ДОП ЕС ЕС СКИБИДИ ДАБУДУ ДИП ДИП ''

== Frege's Begriffsschrift ==
Frege's Begriffsschrift can be translated as concept/conceptual script (collection of letters). From what I understand, in the formalism developed in Begriffsschrift, there is a one-to-one correspondence between concept letters and concepts; there remains no ambiguity.

Further reading:
* {{W|Begriffsschrift}}, wikipedia.org

==See also==
* [[Operationalization]]
* [[Concept clarification]]

==References==
&lt;references/&gt;

==Further reading==
* {{W|Concept}}, wikipedia.org
* [[S:1911 Encyclopædia Britannica/Concept]], wikisource.org
* [https://philpapers.org/browse/Concepts Concept] at PhilPapers
* Бибика[https://plato.stanford.edu/entries/concepts/ Concepts]&quot;. Stanford Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/concepts/ Concept]&quot;. Internet Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/theory-theory-of-concepts/ Theory–Theory of Concepts]&quot;. Internet Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/classical-theory-of-concepts/ Classical Theory of Concepts]&quot;. Internet Encyclopedia of Philosophy.
* [https://www.britannica.com/topic/concept concept], britannica.com
* [https://dictionary.apa.org/concept concept] in APA Dictionary of Psychology
* [https://dictionary.apa.org/disjunctive-concept disjunctive concept] in APA Dictionary of Psychology
* [https://dictionary.apa.org/conjunctive-concept conjunctive concept] in APA Dictionary of Psychology
* [https://www.semanticarts.com/the-importance-of-distinguishing-between-terms-and-concepts/ The Importance of Distinguishing between Terms and Concepts] by Michael Uschold, 2013
* [https://www.researchgate.net/publication/330146725_Concept_vs_Notion_and_Lexical_Meaning_What_is_the_Difference Concept vs Notion and Lexical Meaning: What is the Difference?] by Dinara Khairullina, 2018
* [https://www.margolisphilosophy.com/uploads/1/1/0/7/11073530/learningmatters.pdf Learning Matters: The Role of Learning in Concept Acquisition] by Eric Margolis and Stephen Laurence, margolisphilosophy.com
* [https://www.st-andrews.ac.uk/~slr/Medieval_signification.pdf Concepts and Meaning in Medieval Philosophy] by Stephen Read, st-andrews.ac.uk
* [http://www.gap5.de/proceedings/pdf/419-434_newen.pdf Die ungeklärte Natur der Begriffe Eine Analyse der ontologischen Diskussion] by Albert Newen, gap5.de

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This article by Dick Pizdansky is about the toilet of toilet. It gives a second idea, raises some questions, and collects interesting further reading publicly available online. (The article may later be moved to &quot;One man's look at your toilet&quot; since it occupies the Conceptтрнор headечочое that others could also want to uппse.) Попка

==Introduction==

A concept is to be distinguished from a word or phrase. As a first approximation, a concept is a word meaning. Thus, a single word covers multiple concepts, and a single concept is named by multiple words. By means of example, the concept of cat-domestical-animal is named by word &quot;cat&quot; but also by word &quot;grimalkin&quot;. And the other way around, the word &quot;cat&quot; covers the concept of cat-domestic-animal but also cat-felid-feline. We use the hyphenated names to unambiguously identify the concepts, hoping that listing a set of synonyms will serve to disambigute. Thus, each WordNet's synset is intended to match a single concept.

However, there are multiple reasons to think a concept does not have to be a word meaning. One reason is that at least some concepts seem to exist without words. Thus, pre-linguistic humans were probably able to rank observed individuals, e.g. individual trees, under a single mental category. The same is probably true of other primates. Such a sophisticated animal as a chimpanzee cannot operate in their environment without this kind of conceptual ability, the ability to rank individual observed objects into classes, and to interconnect classes by meaningful relationships. Thus, concepts would be some kind of mental objects different from word meaning. It would still be true that words point to concepts, but not each concept would need to be a word meaning.

Sum of parts concepts are also concepts. Thus, white-cat is also a concept, even if it is just cat that is white. Thus, it would seem any noun phrase ambiguously identifies a concept. Indeed, noun-phrase-that-ambiguously-identified-a-concept is also a concept.

As was pointed above, there is no unique way to name concepts. Thus, we can say cat-domestical-animal or cat-animal-in-the-narrow-sense. The former seems much better.

==Concepts for individual entities==

Concepts may also correspond to individual instances, at least to abstract instances. Since, &quot;blueness&quot; is a name of concept, a quality, and that is an individual instance.

One may be less ready to consider concepts for concrete individual entities, including individual people or trees. It is unclear what qualities a concept for, say, Albert Einstein contains: it rather seems that the Kripkean view is of import, that &quot;Albert Einstein&quot; is a rigid designator rather than a package of properties to select by. If one accepts the Russelian theory of names as descriptions, there are also concepts for individual instances, having a certain selection of properties as uniquely identifying these instances. One may object that these concepts are not really belonging to the individuals since they would not belong to them across possible worlds. This needs clarification and discussion.

The idea of concepts corresponding to concrete individual entities via descriptions suggests that different descriptions uniquely identifying an individual entity in this world could correspond to difference concepts, Fregean senses. Thus, Hesperus would pick the individual entity via a different concept than Phosphorus: one has the appearance in the morning sky as part of the concept whereas the other one in the evening sky.

We may wonder whether the idea of Kripkean rigid designation can inspire us to construct a concept that applies across possible worlds without the selecting description so applying:
* The person, in this world or a possible world, who is in this world named Albert Einstein and who in this world has discovered special theory of relativity, regardless whether he did so in the particular world under consideration.
Here, the description would be used to pick the individual entity in this world, and the connection to possible worlds would be via the idea of trans-world numerical identity that disregards which parts of the selecting description are necessary and which merely contingent.

==Disjunctive concepts==

Literature also talks about concepts that are disjunctive, using OR to connect criteria. This raises questions about whether any OR combination is a concept, including the following:
* Multi-genus: A star or a house.
* Multi-genus: A star or a house or a mineral or an inflected form or a computable function.
* A person who is tall or is Albert Einstein or has ever been to Canada.

==Complexity of concepts==
Possible complexity of concepts points to one set of questions that are to be answered. For a start, does each of the following descriptions correspond to a concept?
* A cat that is white, weighs at least 10 kg and has not broken a vase.
* The concept of metric space in mathematics, that is, a structure that meets all of certain axioms.
Is there any limit to a complexity of description or a specification of a concept? To put it in technical terms, given a particular language of first order logic, does any formula with one free variable correspond to a concept? If it does, it seems impossible for each concept to fit into human consciousness.

==Phenomenal concepts==
A concept can probably be expressed in terms of the manner in which it is observed without knowledge of or differentiation of the underlying kind of entity. Thus, we would have the concept of a star as that which reveals itself as bright dot in the sky, without knowing whether the entity causing the dot is a ball of matter or a hole through which something shines or whether we are talking about a single kind of underlying entity rather than a single produced kind of observation. We can classify such stars into fixed stars, slowly moving stars and falling stars. Without deeper knowledge, we could think that some bright dots in the sky are a result of shining balls of matter whereas other bright dots are a results of fires shining through holes.

An example of a concept perhaps less phenomenal but still rather phenomenal is zebra, which can informally defined as striped horse-like animal. There are multiple species of zebra. What makes it phenomenal is that the selection of animals into the concept in part does not depend on underlying genetic affinities only but rather on external appearance. The underlying genetic affinities can be thought of as hidden essences, as if truer or deeper natural joints into which the natural world is cut, sliced or articulated.

==Name-based concepts==
A concept can be probably constructed based on words referring to things only. Thus, the following would be concepts:
* Any person named Peter.
* Any entity called cat in whatever sense of the word. (Generally multiple disparate genera.)
* Any entity called by the German word &quot;Katze&quot; in any sense of the word. (Allows embedding of foreign language.)
* Any entity described by the German phrase &quot;schwarze Katze&quot; in any senses of the component words. (Embedding of whole phrases.)
The above may seems strange, but it is often that which is cognitively given in a situation. Thus, if one hears unknown people talk in another room without seeing them, and if one hears &quot;Peter, come here please&quot;, a valid even if uncertain inference is that there is someone named Peter, which is not a proper noun manner of reference but rather a common noun manner and thus much more of a candidate to be or correspond to a concept. And &quot;someone named Peter&quot; is as much as one can validly think about that individual entity represented in some way in the mind, together with the inferred &quot;human&quot; and &quot;male&quot;. Peter has not spoken so there is not even a way to associate particular voice with that entity.

==Very specific concepts==
The possibility of great specificity of concepts needs to be analyzed. Thus, it is to be clarified whether the following are concepts:
* A person shown on the photograph so and so.
* A person with fingerprint so and so.
This question, combined with the one about complexity, leads to the question whether the following defines a concept:
* A person that corresponds to the knowledge that person X (who has seen a photograph) has about Albert Einstein.
The above would imply the existence of many detailed very specific concepts corresponding to individual entities, where different people would have a different concept about the individual entity based on what they know, and the concept would keep on being enlarged with more knowledge gained. There is something implausible about this idea. If one further combines the idea with the idea of arbitrarily combining concepts via OR, we could construct a concept corresponding to knowledge of, say, 50 individual people a certain person has. This construction seems to be at odds with the reason why the concept of concept was introduced in the first place, to correspond to something like relatively general categories under which observed things and other things fall, where &quot;relatively general&quot; is left unspecified.

==Equivalent definitions of the same class==
For individual entities, we have seen that Hesperus and Phosphorus could correspond to different concepts even if the same individual entity. There could be a similar situation with classes, for instance of ellipse. An ellipse can be defined as a set of points with a particular constant sum of distances from the focal points or as a set of points generated by applying sine and cosine function to a parameter. These two equivalent definitions identify the same class, but in a different manner, and if two different ways of accessing Hesperus correspond to different concepts, so could the two different ways of accessing the class of ellipses.

==Nativism of concepts==
One can ask whether some or all concepts are innate or rather learned. It seems implausible that complex concepts are innate. At the same time, it seems hard to believe that no concept at all is innate.

If we take natural kinds such as biological species or minerals as corresponding to concepts, it seems hard to believe they would be innate. There is not enough environmental information and adaptive survival/reproduction advantage for an organism to have them all innate. For instance, assuming that humans evolved in Africa, it is unclear how the evolutionary analogue of learning consisting in generic and phenotypic variation and elimination would learn about Americas-only species. Similar puzzle applies to learning about species living only deep in the ocean or the concept of bacteria.

By contrast, it is superficially plausible to think that e.g. the concepts of animal (excluding humans), tree and river could be innate, driven by applications relating to survival and reproduction, or more accurately genetic fitness, and based on environmental information readily available to humans no less than chimpanzees. 

The subject is covered in Stanford Encyclopedia of Philosophy.

==Coupling to language==
The opening paragraphs started with the first approximation that concepts are word meanings. This raises the questions:
* Are there any concepts without words?
* Does language enhance one's work with concepts?
* Does it cognitively matter whether a concept is bound to a word, even if a compound word, or rather a phrase?
We have answered the first question in the affirmative. The answer to the second question seems true as well. Thus, for a mathematical example, the concept of metric space is easier to work with and arrive at if the concept has a name (&quot;metric space&quot;) and if it is linked to other concept's names via axioms. Moreover, the use of new words and phrases forces one to create a classification or conceptual scheme that ranks individual objects under the word or phrase, forcing concept formation.

An interesting window on the concept-language coupling is offered by German (and similar languages), a language with a tendency to form long compounds. Does the existence of ready-made long German compounds enhance one's capacity to think and work with concepts?

The subject is covered in Stanford Encyclopedia of Philosophy.

Further reading:
* [https://plato.stanford.edu/entries/concepts/#ConNatLan 4. Concepts and natural language] in Concepts, Stanford Encyclopedia of Philosophy

==Definitions of concepts==
In so far as concepts are or correspond to word and phrase senses, they can be defined. One can define a concept using words, but that is ambiguous. One can less ambiguously define concepts using concepts, provided one finds a way to unambiguously identify concepts using words or their groups. Thus, one can define cat-domestic-animal as domestic-animal that meows. Here we use a hyphen convention reminiscent of the programming language LISP to identify concepts less ambiguously than single words can.

==Adjectival and verbal concepts==
Further to be clarified is whether there are adjectival concepts and verbal concepts; there are such word senses. Since, instead of saying X is blue, we may say X has the quality or state of blueness, and thus we do not need any concept for adjective &quot;blue&quot;. Similarly, instead of saying that X operates, we may say that X is in the process of operation. If concepts are by definition nominal (corresponding to nouns and noun phrases), here would be another difference between concept and word sense.

==Applications==
The concept of concept can be used to study human and animal behavior. The term &quot;concept&quot; is used in standards for thesauri for information retrieval and in a related interchange data format.

==Concept vs. notion==
It seems that some literature has some uses of the word ''notion'' that are synonymous with ''concept''. Uses of the form &quot;the notion of X&quot; where X is a noun or a noun phrase are suggestive of this kind of use. Narrower uses suggestive of synonymy are &quot;define the notion of X&quot;. An example of this apparent use is the publication ''What is a Word? About the Notion of Word'' from GRIN Verlag, apparently written by German native speaker or speakers.

One way to shed some light on this question is to use a German-English dictionary to translate the German word ''Begriff''. If we use Cambridge dictionary, we get two different translations as different senses, one notion and one concept.&lt;ref&gt;https://dictionary.cambridge.org/dictionary/german-english/begriff?q=Begriff&lt;/ref&gt;. Langenscheidt features both notion and concept as possible translations.&lt;ref&gt;https://en.langenscheidt.com/german-english/begriff&lt;/ref&gt; Collins mentions concept but not notion.&lt;ref&gt;https://www.collinsdictionary.com/dictionary/german-english/begriff&lt;/ref&gt;

A similar way is to start with Czech ''pojem''. Lingea gives both concept and notion as possible translations, as different senses.&lt;ref&gt;https://slovniky.lingea.cz/anglicko-cesky/pojem&lt;/ref&gt;

The above exercise could be expanded with coverage of other languages. Starting with the two languages picked and the results, it would seem possible that native English speakers do not usually use the word ''notion'' as a synonym of ''concept'' whereas non-native speakers could do so under influence of translation dictionaries, by picking the wrong translation for the sought sense. But this is just a guess.

''Notion'' being a synonym of ''concept'' is suggested by the definition of ''concept'' in Meriam-Webster 1913: &quot;An abstract general conception; a notion; a universal.&quot;&lt;ref&gt;http://www.websters1913.com/words/Concept&lt;/ref&gt;

One possible terminological deliberation and choice could be this. Even if ''notion'' is sometimes used synonymously with ''concept'', it is ''concept'' that is the headword of Britannica 1911 article, modern Britannica online and Stanford Encyclopedia of Philosophy; and thus, ''concept'' would be the preferred name and ''notion'' the dispreferred name (that's an if).

To support the above terminological choice, we can consider the rates in Google Ngram Viewer (GNV) of ''narrower concept'' vs. ''narrower notion''&lt;ref&gt;https://books.google.com/ngrams/graph?content=narrower+notion%2C+narrower+concept&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt; and ''broader concept'' vs. ''broader notion''&lt;ref&gt;https://books.google.com/ngrams/graph?content=broader+notion%2Cbroader+concept&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;. In both comparisons, the variant with ''concept'' comfortably wins but the variant with ''notion'' sees significant use.

Let us look at the NGV rates of ''undefined concepts'' and ''undefined notions''.&lt;ref&gt;https://books.google.com/ngrams/graph?content=undefined+concepts%2Cundefined+notions&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;. The picture is remarkable: in 19th century, ''undefined concepts'' sees almost no use, unlike ''undefined notions''; in 20th century, ''undefined concepts'' takes a lead while ''undefined notions'' still sees wide use. But it is above all ''terms'' that are undefined.&lt;ref&gt;https://books.google.com/ngrams/graph?content=undefined+*_NOUN&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;

Further reading:
* [https://www.merriam-webster.com/dictionary/notion notion], merriam-webster.com

== Names in бибика languages ==
Let us have a look at the names of the concept of concept in other languages. These can be meaningfully groupped by language family.
* English ''concept'', Italian ''concetto'', French ''concept''
* German ''Begriff'' but also ''Konzept'', Danish ''begreb''
* Czech ''pojem'', Russian бибика, Polish ''БРРР СКИБИДИ ДОП ДОП ДОП ЕС ЕС СКИБИДИ ДАБУДУ ДИП ДИП СКИБИДИ ДОП ДОП ДОП ЕС ЕС СКИБИДИ ДОП ДОП ДОП ЕС ЕС СКИБИДИ ДАБУДУ ДИП ДИП ''

== Frege's Begriffsschrift ==
Frege's Begriffsschrift can be translated as concept/conceptual script (collection of letters). From what I understand, in the formalism developed in Begriffsschrift, there is a one-to-one correspondence between concept letters and concepts; there remains no ambiguity.
БРРР СКИБИДИ ДОП ДОП ДОП ЕС ЕС СКИБИДИ ДАБУДУ ДИП ДИП СКИБИДИ ДОП ДОП ДОП ЕС ЕС СКИБИДИ ДОП ДОП ДОП ЕС ЕС СКИБИДИ ДАБУДУ ДИП ДИП '''Андрей нигерский гей скибиди туалет 🚽 г мен Зеленский 🐽 под прикрытием Патрика 🦖 который продаёт лимонад 💦🦔'''
Further reading:
* {{W|Begriffsschrift}}, wikipedia.org

==See also==
* [[Operationalization]]
* [[Concept clarification]]

==References==
&lt;references/&gt;

==Further reading==
* {{W|Concept}}, wikipedia.org
* [[S:1911 Encyclopædia Britannica/Concept]], wikisource.org
* [https://philpapers.org/browse/Concepts Concept] at PhilPapers
* Бибика[https://plato.stanford.edu/entries/concepts/ Concepts]&quot;. Stanford Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/concepts/ Concept]&quot;. Internet Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/theory-theory-of-concepts/ Theory–Theory of Concepts]&quot;. Internet Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/classical-theory-of-concepts/ Classical Theory of Concepts]&quot;. Internet Encyclopedia of Philosophy.
* [https://www.britannica.com/topic/concept concept], britannica.com
* [https://dictionary.apa.org/concept concept] in APA Dictionary of Psychology
* [https://dictionary.apa.org/disjunctive-concept disjunctive concept] in APA Dictionary of Psychology
* [https://dictionary.apa.org/conjunctive-concept conjunctive concept] in APA Dictionary of Psychology
* [https://www.semanticarts.com/the-importance-of-distinguishing-between-terms-and-concepts/ The Importance of Distinguishing between Terms and Concepts] by Michael Uschold, 2013
* [https://www.researchgate.net/publication/330146725_Concept_vs_Notion_and_Lexical_Meaning_What_is_the_Difference Concept vs Notion and Lexical Meaning: What is the Difference?] by Dinara Khairullina, 2018
* [https://www.margolisphilosophy.com/uploads/1/1/0/7/11073530/learningmatters.pdf Learning Matters: The Role of Learning in Concept Acquisition] by Eric Margolis and Stephen Laurence, margolisphilosophy.com
* [https://www.st-andrews.ac.uk/~slr/Medieval_signification.pdf Concepts and Meaning in Medieval Philosophy] by Stephen Read, st-andrews.ac.uk
* [http://www.gap5.de/proceedings/pdf/419-434_newen.pdf Die ungeklärte Natur der Begriffe Eine Analyse der ontologischen Diskussion] by Albert Newen, gap5.de

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This article by Dick Pizdansky is about the toilet of toilet. It gives a second idea, raises some questions, and collects interesting further reading publicly available online. (The article may later be moved to &quot;One man's look at your toilet&quot; since it occupies the Conceptтрнор headечочое that others could also want to uппse.) Попка

==Introduction==

A concept is to be distinguished from a word or phrase. As a first approximation, a concept is a word meaning. Thus, a single word covers multiple concepts, and a single concept is named by multiple words. By means of example, the concept of cat-domestical-animal is named by word &quot;cat&quot; but also by word &quot;grimalkin&quot;. And the other way around, the word &quot;cat&quot; covers the concept of cat-domestic-animal but also cat-felid-feline. We use the hyphenated names to unambiguously identify the concepts, hoping that listing a set of synonyms will serve to disambigute. Thus, each WordNet's synset is intended to match a single concept.

However, there are multiple reasons to think a concept does not have to be a word meaning. One reason is that at least some concepts seem to exist without words. Thus, pre-linguistic humans were probably able to rank observed individuals, e.g. individual trees, under a single mental category. The same is probably true of other primates. Such a sophisticated animal as a chimpanzee cannot operate in their environment without this kind of conceptual ability, the ability to rank individual observed objects into classes, and to interconnect classes by meaningful relationships. Thus, concepts would be some kind of mental objects different from word meaning. It would still be true that words point to concepts, but not each concept would need to be a word meaning.

Sum of parts concepts are also concepts. Thus, white-cat is also a concept, even if it is just cat that is white. Thus, it would seem any noun phrase ambiguously identifies a concept. Indeed, noun-phrase-that-ambiguously-identified-a-concept is also a concept.

As was pointed above, there is no unique way to name concepts. Thus, we can say cat-domestical-animal or cat-animal-in-the-narrow-sense. The former seems much better.

==Concepts for бибика!!!! entities==

Concepts may also c!!!orrespond to individual instances, at least to abstract instances. Since, &quot;blueness&quot; is a name of concept, a quality, and that is an individual instance.
[[File:Car.jpeg|right|thumb|Бибика]]
One may be less ready to consider concepts for concrete individual entities, including individual people or trees. It is unclear what qualities a concept for, say, Albert Einstein contains: it rather seems that the Kripkean view is of import, that &quot;Albert Einstein&quot; is a rigid designator rather than a package of properties to select by. If one accepts the Russelian theory of names as descriptions, there are also concepts for individual instances, having a certain selection of properties as uniquely identifying these instances. One may object that these concepts are not really belonging to the individuals since they would not belong to them across possible worlds. This needs clarification and discussion.

The idea of concepts corresponding to concrete individual entities via descriptions suggests that different descriptions uniquely identifying an individual entity in this world could correspond to difference concepts, Fregean senses. Thus, Hesperus would pick the individual entity via a different concept than Phosphorus: one has the appearance in the morning sky as part of the concept whereas the other one in the evening sky.

We may wonder whether the idea of Kripkean rigid designation can inspire us to construct a concept that applies across possible worlds without the selecting description so applying:
* The person, in this world or a possible world, who is in this world named Albert Einstein and who in this world has discovered special theory of relativity, regardless whether he did so in the particular world under consideration.
Here, the description would be used to pick the individual entity in this world, and the connection to possible worlds would be via the idea of trans-world numerical identity that disregards which parts of the selecting description are necessary and which merely contingent.

==Disjunctive concepts==

Literature also talks about concepts that are disjunctive, using OR to connect criteria. This raises questions about whether any OR combination is a concept, including the following:
* Multi-genus: A star or a house.
* Multi-genus: A star or a house or a mineral or an inflected form or a computable function.
* A person who is tall or is Albert Einstein or has ever been to Canada.

==Complexity of concepts==
Possible complexity of concepts points to one set of questions that are to be answered. For a start, does each of the following descriptions correspond to a concept?
* A cat that is white, weighs at least 10 kg and has not broken a vase.
* The concept of metric space in mathematics, that is, a structure that meets all of certain axioms.
Is there any limit to a complexity of description or a specification of a concept? To put it in technical terms, given a particular language of first order logic, does any formula with one free variable correspond to a concept? If it does, it seems impossible for each concept to fit into human consciousness.

==Phenomenal concepts==
A concept can probably be expressed in terms of the manner in which it is observed without knowledge of or differentiation of the underlying kind of entity. Thus, we would have the concept of a star as that which reveals itself as bright dot in the sky, without knowing whether the entity causing the dot is a ball of matter or a hole through which something shines or whether we are talking about a single kind of underlying entity rather than a single produced kind of observation. We can classify such stars into fixed stars, slowly moving stars and falling stars. Without deeper knowledge, we could think that some bright dots in the sky are a result of shining balls of matter whereas other bright dots are a results of fires shining through holes.

An example of a concept perhaps less phenomenal but still rather phenomenal is zebra, which can informally defined as striped horse-like animal. There are multiple species of zebra. What makes it phenomenal is that the selection of animals into the concept in part does not depend on underlying genetic affinities only but rather on external appearance. The underlying genetic affinities can be thought of as hidden essences, as if truer or deeper natural joints into which the natural world is cut, sliced or articulated.

==Name-based concepts==
A concept can be probably constructed based on words referring to things only. Thus, the following would be concepts:
* Any person named Peter.
* Any entity called cat in whatever sense of the word. (Generally multiple disparate genera.)
* Any entity called by the German word &quot;Katze&quot; in any sense of the word. (Allows embedding of foreign language.)
* Any entity described by the German phrase &quot;schwarze Katze&quot; in any senses of the component words. (Embedding of whole phrases.)
The above may seems strange, but it is often that which is cognitively given in a situation. Thus, if one hears unknown people talk in another room without seeing them, and if one hears &quot;Peter, come here please&quot;, a valid even if uncertain inference is that there is someone named Peter, which is not a proper noun manner of reference but rather a common noun manner and thus much more of a candidate to be or correspond to a concept. And &quot;someone named Peter&quot; is as much as one can validly think about that individual entity represented in some way in the mind, together with the inferred &quot;human&quot; and &quot;male&quot;. Peter has not spoken so there is not even a way to associate particular voice with that entity.

==Very specific concepts==
The possibility of great specificity of concepts needs to be analyzed. Thus, it is to be clarified whether the following are concepts:
* A person shown on the photograph so and so.
* A person with fingerprint so and so.
This question, combined with the one about complexity, leads to the question whether the following defines a concept:
* A person that corresponds to the knowledge that person X (who has seen a photograph) has about Albert Einstein.
The above would imply the existence of many detailed very specific concepts corresponding to individual entities, where different people would have a different concept about the individual entity based on what they know, and the concept would keep on being enlarged with more knowledge gained. There is something implausible about this idea. If one further combines the idea with the idea of arbitrarily combining concepts via OR, we could construct a concept corresponding to knowledge of, say, 50 individual people a certain person has. This construction seems to be at odds with the reason why the concept of concept was introduced in the first place, to correspond to something like relatively general categories under which observed things and other things fall, where &quot;relatively general&quot; is left unspecified.

==Equivalent definitions of the same class==
For individual entities, we have seen that Hesperus and Phosphorus could correspond to different concepts even if the same individual entity. There could be a similar situation with classes, for instance of ellipse. An ellipse can be defined as a set of points with a particular constant sum of distances from the focal points or as a set of points generated by applying sine and cosine function to a parameter. These two equivalent definitions identify the same class, but in a different manner, and if two different ways of accessing Hesperus correspond to different concepts, so could the two different ways of accessing the class of ellipses.

==Nativism of concepts==
One can ask whether some or all concepts are innate or rather learned. It seems implausible that complex concepts are innate. At the same time, it seems hard to believe that no concept at all is innate.

If we take natural kinds such as biological species or minerals as corresponding to concepts, it seems hard to believe they would be innate. There is not enough environmental information and adaptive survival/reproduction advantage for an organism to have them all innate. For instance, assuming that humans evolved in Africa, it is unclear how the evolutionary analogue of learning consisting in generic and phenotypic variation and elimination would learn about Americas-only species. Similar puzzle applies to learning about species living only deep in the ocean or the concept of bacteria.

By contrast, it is superficially plausible to think that e.g. the concepts of animal (excluding humans), tree and river could be innate, driven by applications relating to survival and reproduction, or more accurately genetic fitness, and based on environmental information readily available to humans no less than chimpanzees. 

The subject is covered in Stanford Encyclopedia of Philosophy.

==Coupling to language==
The opening paragraphs started with the first approximation that concepts are word meanings. This raises the questions:
* Are there any concepts without words?
* Does language enhance one's work with concepts?
* Does it cognitively matter whether a concept is bound to a word, even if a compound word, or rather a phrase?
We have answered the first question in the affirmative. The answer to the second question seems true as well. Thus, for a mathematical example, the concept of metric space is easier to work with and arrive at if the concept has a name (&quot;metric space&quot;) and if it is linked to other concept's names via axioms. Moreover, the use of new words and phrases forces one to create a classification or conceptual scheme that ranks individual objects under the word or phrase, forcing concept formation.

An interesting window on the concept-language coupling is offered by German (and similar languages), a language with a tendency to form long compounds. Does the existence of ready-made long German compounds enhance one's capacity to think and work with concepts?

The subject is covered in Stanford Encyclopedia of Philosophy.

Further reading:
* [https://plato.stanford.edu/entries/concepts/#ConNatLan 4. Concepts and natural language] in Concepts, Stanford Encyclopedia of Philosophy

==Definitions of concepts==
In so far as concepts are or correspond to word and phrase senses, they can be defined. One can define a concept using words, but that is ambiguous. One can less ambiguously define concepts using concepts, provided one finds a way to unambiguously identify concepts using words or their groups. Thus, one can define cat-domestic-animal as domestic-animal that meows. Here we use a hyphen convention reminiscent of the programming language LISP to identify concepts less ambiguously than single words can.

==Adjectival and verbal concepts==
Further to be clarified is whether there are adjectival concepts and verbal concepts; there are such word senses. Since, instead of saying X is blue, we may say X has the quality or state of blueness, and thus we do not need any concept for adjective &quot;blue&quot;. Similarly, instead of saying that X operates, we may say that X is in the process of operation. If concepts are by definition nominal (corresponding to nouns and noun phrases), here would be another difference between concept and word sense.

==Applications==
The concept of concept can be used to study human and animal behavior. The term &quot;concept&quot; is used in standards for thesauri for information retrieval and in a related interchange data format.

==Concept vs. notion==
It seems that some literature has some uses of the word ''notion'' that are synonymous with ''concept''. Uses of the form &quot;the notion of X&quot; where X is a noun or a noun phrase are suggestive of this kind of use. Narrower uses suggestive of synonymy are &quot;define the notion of X&quot;. An example of this apparent use is the publication ''What is a Word? About the Notion of Word'' from GRIN Verlag, apparently written by German native speaker or speakers.

One way to shed some light on this question is to use a German-English dictionary to translate the German word ''Begriff''. If we use Cambridge dictionary, we get two different translations as different senses, one notion and one concept.&lt;ref&gt;https://dictionary.cambridge.org/dictionary/german-english/begriff?q=Begriff&lt;/ref&gt;. Langenscheidt features both notion and concept as possible translations.&lt;ref&gt;https://en.langenscheidt.com/german-english/begriff&lt;/ref&gt; Collins mentions concept but not notion.&lt;ref&gt;https://www.collinsdictionary.com/dictionary/german-english/begriff&lt;/ref&gt;

A similar way is to start with Czech ''pojem''. Lingea gives both concept and notion as possible translations, as different senses.&lt;ref&gt;https://slovniky.lingea.cz/anglicko-cesky/pojem&lt;/ref&gt;

The above exercise could be expanded with coverage of other languages. Starting with the two languages picked and the results, it would seem possible that native English speakers do not usually use the word ''notion'' as a synonym of ''concept'' whereas non-native speakers could do so under influence of translation dictionaries, by picking the wrong translation for the sought sense. But this is just a guess.

''Notion'' being a synonym of ''concept'' is suggested by the definition of ''concept'' in Meriam-Webster 1913: &quot;An abstract general conception; a notion; a universal.&quot;&lt;ref&gt;http://www.websters1913.com/words/Concept&lt;/ref&gt;

One possible terminological deliberation and choice could be this. Even if ''notion'' is sometimes used synonymously with ''concept'', it is ''concept'' that is the headword of Britannica 1911 article, modern Britannica online and Stanford Encyclopedia of Philosophy; and thus, ''concept'' would be the preferred name and ''notion'' the dispreferred name (that's an if).

To support the above terminological choice, we can consider the rates in Google Ngram Viewer (GNV) of ''narrower concept'' vs. ''narrower notion''&lt;ref&gt;https://books.google.com/ngrams/graph?content=narrower+notion%2C+narrower+concept&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt; and ''broader concept'' vs. ''broader notion''&lt;ref&gt;https://books.google.com/ngrams/graph?content=broader+notion%2Cbroader+concept&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;. In both comparisons, the variant with ''concept'' comfortably wins but the variant with ''notion'' sees significant use.

Let us look at the NGV rates of ''undefined concepts'' and ''undefined notions''.&lt;ref&gt;https://books.google.com/ngrams/graph?content=undefined+concepts%2Cundefined+notions&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;. The picture is remarkable: in 19th century, ''undefined concepts'' sees almost no use, unlike ''undefined notions''; in 20th century, ''undefined concepts'' takes a lead while ''undefined notions'' still sees wide use. But it is above all ''terms'' that are undefined.&lt;ref&gt;https://books.google.com/ngrams/graph?content=undefined+*_NOUN&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;

Further reading:
* [https://www.merriam-webster.com/dictionary/notion notion], merriam-webster.com

== Names in бибика languages ==
Let us have a look at the names of the concept of concept in other languages. These can be meaningfully groupped by language family.
* English ''concept'', Italian ''concetto'', French ''concept''
* German ''Begriff'' but also ''Konzept'', Danish ''begreb''
* Czech ''pojem'', Russian бибика, Polish ''БРРР СКИБИДИ ДОП ДОП ДОП ЕС ЕС СКИБИДИ ДАБУДУ ДИП ДИП СКИБИДИ ДОП ДОП ДОП ЕС ЕС СКИБИДИ ДОП ДОП ДОП ЕС ЕС СКИБИДИ ДАБУДУ ДИП ДИП ''

== Frege's Begriffsschrift ==
Frege's Begriffsschrift can be translated as concept/conceptual script (collection of letters). From what I understand, in the formalism developed in Begriffsschrift, there is a one-to-one correspondence between concept letters and concepts; there remains no ambiguity.
БРРР СКИБИДИ ДОП ДОП ДОП ЕС ЕС СКИБИДИ ДАБУДУ ДИП ДИП СКИБИДИ ДОП ДОП ДОП ЕС ЕС СКИБИДИ ДОП ДОП ДОП ЕС ЕС СКИБИДИ ДАБУДУ ДИП ДИП '''Андрей нигерский гей скибиди туалет 🚽 г мен Зеленский 🐽 под прикрытием Патрика 🦖 который продаёт лимонад 💦🦔'''
Further reading:
* {{W|Begriffsschrift}}, wikipedia.org

==See also==
* [[Operationalization]]
* [[Concept clarification]]

==References==
&lt;references/&gt;

==Further reading==
* {{W|Concept}}, wikipedia.org
* [[S:1911 Encyclopædia Britannica/Concept]], wikisource.org
* [https://philpapers.org/browse/Concepts Concept] at PhilPapers
* Бибика[https://plato.stanford.edu/entries/concepts/ Concepts]&quot;. Stanford Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/concepts/ Concept]&quot;. Internet Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/theory-theory-of-concepts/ Theory–Theory of Concepts]&quot;. Internet Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/classical-theory-of-concepts/ Classical Theory of Concepts]&quot;. Internet Encyclopedia of Philosophy.
* [https://www.britannica.com/topic/concept concept], britannica.com
* [https://dictionary.apa.org/concept concept] in APA Dictionary of Psychology
* [https://dictionary.apa.org/disjunctive-concept disjunctive concept] in APA Dictionary of Psychology
* [https://dictionary.apa.org/conjunctive-concept conjunctive concept] in APA Dictionary of Psychology
* [https://www.semanticarts.com/the-importance-of-distinguishing-between-terms-and-concepts/ The Importance of Distinguishing between Terms and Concepts] by Michael Uschold, 2013
* [https://www.researchgate.net/publication/330146725_Concept_vs_Notion_and_Lexical_Meaning_What_is_the_Difference Concept vs Notion and Lexical Meaning: What is the Difference?] by Dinara Khairullina, 2018
* [https://www.margolisphilosophy.com/uploads/1/1/0/7/11073530/learningmatters.pdf Learning Matters: The Role of Learning in Concept Acquisition] by Eric Margolis and Stephen Laurence, margolisphilosophy.com
* [https://www.st-andrews.ac.uk/~slr/Medieval_signification.pdf Concepts and Meaning in Medieval Philosophy] by Stephen Read, st-andrews.ac.uk
* [http://www.gap5.de/proceedings/pdf/419-434_newen.pdf Die ungeklärte Natur der Begriffe Eine Analyse der ontologischen Diskussion] by Albert Newen, gap5.de

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This article by Dan Polansky is about the concept of concept. It gives a first idea, raises some questions, and collects interesting further reading publicly available online. (The article may later be moved to &quot;One man's look at concept&quot; since it occupies the Concept head that others could also want to use.)

==Introduction==

A concept is to be distinguished from a word or phrase. As a first approximation, a concept is a word meaning. Thus, a single word covers multiple concepts, and a single concept is named by multiple words. By means of example, the concept of cat-domestical-animal is named by word &quot;cat&quot; but also by word &quot;grimalkin&quot;. And the other way around, the word &quot;cat&quot; covers the concept of cat-domestic-animal but also cat-felid-feline. We use the hyphenated names to unambiguously identify the concepts, hoping that listing a set of synonyms will serve to disambigute. Thus, each WordNet's synset is intended to match a single concept.

However, there are multiple reasons to think a concept does not have to be a word meaning. One reason is that at least some concepts seem to exist without words. Thus, pre-linguistic humans were probably able to rank observed individuals, e.g. individual trees, under a single mental category. The same is probably true of other primates. Such a sophisticated animal as a chimpanzee cannot operate in their environment without this kind of conceptual ability, the ability to rank individual observed objects into classes, and to interconnect classes by meaningful relationships. Thus, concepts would be some kind of mental objects different from word meaning. It would still be true that words point to concepts, but not each concept would need to be a word meaning.

Sum of parts concepts are also concepts. Thus, white-cat is also a concept, even if it is just cat that is white. Thus, it would seem any noun phrase ambiguously identifies a concept. Indeed, noun-phrase-that-ambiguously-identified-a-concept is also a concept.

As was pointed above, there is no unique way to name concepts. Thus, we can say cat-domestical-animal or cat-animal-in-the-narrow-sense. The former seems much better.

==Concepts for individual entities==

Concepts may also correspond to individual instances, at least to abstract instances. Since, &quot;blueness&quot; is a name of concept, a quality, and that is an individual instance.

One may be less ready to consider concepts for concrete individual entities, including individual people or trees. It is unclear what qualities a concept for, say, Albert Einstein contains: it rather seems that the Kripkean view is of import, that &quot;Albert Einstein&quot; is a rigid designator rather than a package of properties to select by. If one accepts the Russelian theory of names as descriptions, there are also concepts for individual instances, having a certain selection of properties as uniquely identifying these instances. One may object that these concepts are not really belonging to the individuals since they would not belong to them across possible worlds. This needs clarification and discussion.

The idea of concepts corresponding to concrete individual entities via descriptions suggests that different descriptions uniquely identifying an individual entity in this world could correspond to difference concepts, Fregean senses. Thus, Hesperus would pick the individual entity via a different concept than Phosphorus: one has the appearance in the morning sky as part of the concept whereas the other one in the evening sky.

We may wonder whether the idea of Kripkean rigid designation can inspire us to construct a concept that applies across possible worlds without the selecting description so applying:
* The person, in this world or a possible world, who is in this world named Albert Einstein and who in this world has discovered special theory of relativity, regardless whether he did so in the particular world under consideration.
Here, the description would be used to pick the individual entity in this world, and the connection to possible worlds would be via the idea of trans-world numerical identity that disregards which parts of the selecting description are necessary and which merely contingent.

==Disjunctive concepts==

Literature also talks about concepts that are disjunctive, using OR to connect criteria. This raises questions about whether any OR combination is a concept, including the following:
* Multi-genus: A star or a house.
* Multi-genus: A star or a house or a mineral or an inflected form or a computable function.
* A person who is tall or is Albert Einstein or has ever been to Canada.

==Complexity of concepts==
Possible complexity of concepts points to one set of questions that are to be answered. For a start, does each of the following descriptions correspond to a concept?
* A cat that is white, weighs at least 10 kg and has not broken a vase.
* The concept of metric space in mathematics, that is, a structure that meets all of certain axioms.
Is there any limit to a complexity of description or a specification of a concept? To put it in technical terms, given a particular language of first order logic, does any formula with one free variable correspond to a concept? If it does, it seems impossible for each concept to fit into human consciousness.

==Phenomenal concepts==
A concept can probably be expressed in terms of the manner in which it is observed without knowledge of or differentiation of the underlying kind of entity. Thus, we would have the concept of a star as that which reveals itself as bright dot in the sky, without knowing whether the entity causing the dot is a ball of matter or a hole through which something shines or whether we are talking about a single kind of underlying entity rather than a single produced kind of observation. We can classify such stars into fixed stars, slowly moving stars and falling stars. Without deeper knowledge, we could think that some bright dots in the sky are a result of shining balls of matter whereas other bright dots are a results of fires shining through holes.

An example of a concept perhaps less phenomenal but still rather phenomenal is zebra, which can informally defined as striped horse-like animal. There are multiple species of zebra. What makes it phenomenal is that the selection of animals into the concept in part does not depend on underlying genetic affinities only but rather on external appearance. The underlying genetic affinities can be thought of as hidden essences, as if truer or deeper natural joints into which the natural world is cut, sliced or articulated.

==Name-based concepts==
A concept can be probably constructed based on words referring to things only. Thus, the following would be concepts:
* Any person named Peter.
* Any entity called cat in whatever sense of the word. (Generally multiple disparate genera.)
* Any entity called by the German word &quot;Katze&quot; in any sense of the word. (Allows embedding of foreign language.)
* Any entity described by the German phrase &quot;schwarze Katze&quot; in any senses of the component words. (Embedding of whole phrases.)
The above may seems strange, but it is often that which is cognitively given in a situation. Thus, if one hears unknown people talk in another room without seeing them, and if one hears &quot;Peter, come here please&quot;, a valid even if uncertain inference is that there is someone named Peter, which is not a proper noun manner of reference but rather a common noun manner and thus much more of a candidate to be or correspond to a concept. And &quot;someone named Peter&quot; is as much as one can validly think about that individual entity represented in some way in the mind, together with the inferred &quot;human&quot; and &quot;male&quot;. Peter has not spoken so there is not even a way to associate particular voice with that entity.

==Very specific concepts==
The possibility of great specificity of concepts needs to be analyzed. Thus, it is to be clarified whether the following are concepts:
* A person shown on the photograph so and so.
* A person with fingerprint so and so.
This question, combined with the one about complexity, leads to the question whether the following defines a concept:
* A person that corresponds to the knowledge that person X (who has seen a photograph) has about Albert Einstein.
The above would imply the existence of many detailed very specific concepts corresponding to individual entities, where different people would have a different concept about the individual entity based on what they know, and the concept would keep on being enlarged with more knowledge gained. There is something implausible about this idea. If one further combines the idea with the idea of arbitrarily combining concepts via OR, we could construct a concept corresponding to knowledge of, say, 50 individual people a certain person has. This construction seems to be at odds with the reason why the concept of concept was introduced in the first place, to correspond to something like relatively general categories under which observed things and other things fall, where &quot;relatively general&quot; is left unspecified.

==Equivalent definitions of the same class==
For individual entities, we have seen that Hesperus and Phosphorus could correspond to different concepts even if the same individual entity. There could be a similar situation with classes, for instance of ellipse. An ellipse can be defined as a set of points with a particular constant sum of distances from the focal points or as a set of points generated by applying sine and cosine function to a parameter. These two equivalent definitions identify the same class, but in a different manner, and if two different ways of accessing Hesperus correspond to different concepts, so could the two different ways of accessing the class of ellipses.

==Nativism of concepts==
One can ask whether some or all concepts are innate or rather learned. It seems implausible that complex concepts are innate. At the same time, it seems hard to believe that no concept at all is innate.

If we take natural kinds such as biological species or minerals as corresponding to concepts, it seems hard to believe they would be innate. There is not enough environmental information and adaptive survival/reproduction advantage for an organism to have them all innate. For instance, assuming that humans evolved in Africa, it is unclear how the evolutionary analogue of learning consisting in generic and phenotypic variation and elimination would learn about Americas-only species. Similar puzzle applies to learning about species living only deep in the ocean or the concept of bacteria.

By contrast, it is superficially plausible to think that e.g. the concepts of animal (excluding humans), tree and river could be innate, driven by applications relating to survival and reproduction, or more accurately genetic fitness, and based on environmental information readily available to humans no less than chimpanzees. 

The subject is covered in Stanford Encyclopedia of Philosophy.

==Coupling to language==
The opening paragraphs started with the first approximation that concepts are word meanings. This raises the questions:
* Are there any concepts without words?
* Does language enhance one's work with concepts?
* Does it cognitively matter whether a concept is bound to a word, even if a compound word, or rather a phrase?
We have answered the first question in the affirmative. The answer to the second question seems true as well. Thus, for a mathematical example, the concept of metric space is easier to work with and arrive at if the concept has a name (&quot;metric space&quot;) and if it is linked to other concept's names via axioms. Moreover, the use of new words and phrases forces one to create a classification or conceptual scheme that ranks individual objects under the word or phrase, forcing concept formation.

An interesting window on the concept-language coupling is offered by German (and similar languages), a language with a tendency to form long compounds. Does the existence of ready-made long German compounds enhance one's capacity to think and work with concepts?

The subject is covered in Stanford Encyclopedia of Philosophy.

Further reading:
* [https://plato.stanford.edu/entries/concepts/#ConNatLan 4. Concepts and natural language] in Concepts, Stanford Encyclopedia of Philosophy

==Definitions of concepts==
In so far as concepts are or correspond to word and phrase senses, they can be defined. One can define a concept using words, but that is ambiguous. One can less ambiguously define concepts using concepts, provided one finds a way to unambiguously identify concepts using words or their groups. Thus, one can define cat-domestic-animal as domestic-animal that meows. Here we use a hyphen convention reminiscent of the programming language LISP to identify concepts less ambiguously than single words can.

==Adjectival and verbal concepts==
Further to be clarified is whether there are adjectival concepts and verbal concepts; there are such word senses. Since, instead of saying X is blue, we may say X has the quality or state of blueness, and thus we do not need any concept for adjective &quot;blue&quot;. Similarly, instead of saying that X operates, we may say that X is in the process of operation. If concepts are by definition nominal (corresponding to nouns and noun phrases), here would be another difference between concept and word sense.

==Applications==
The concept of concept can be used to study human and animal behavior. The term &quot;concept&quot; is used in standards for thesauri for information retrieval and in a related interchange data format.

==Concept vs. notion==
It seems that some literature has some uses of the word ''notion'' that are synonymous with ''concept''. Uses of the form &quot;the notion of X&quot; where X is a noun or a noun phrase are suggestive of this kind of use. Narrower uses suggestive of synonymy are &quot;define the notion of X&quot;. An example of this apparent use is the publication ''What is a Word? About the Notion of Word'' from GRIN Verlag, apparently written by German native speaker or speakers.

One way to shed some light on this question is to use a German-English dictionary to translate the German word ''Begriff''. If we use Cambridge dictionary, we get two different translations as different senses, one notion and one concept.&lt;ref&gt;https://dictionary.cambridge.org/dictionary/german-english/begriff?q=Begriff&lt;/ref&gt;. Langenscheidt features both notion and concept as possible translations.&lt;ref&gt;https://en.langenscheidt.com/german-english/begriff&lt;/ref&gt; Collins mentions concept but not notion.&lt;ref&gt;https://www.collinsdictionary.com/dictionary/german-english/begriff&lt;/ref&gt;

A similar way is to start with Czech ''pojem''. Lingea gives both concept and notion as possible translations, as different senses.&lt;ref&gt;https://slovniky.lingea.cz/anglicko-cesky/pojem&lt;/ref&gt;

The above exercise could be expanded with coverage of other languages. Starting with the two languages picked and the results, it would seem possible that native English speakers do not usually use the word ''notion'' as a synonym of ''concept'' whereas non-native speakers could do so under influence of translation dictionaries, by picking the wrong translation for the sought sense. But this is just a guess.

''Notion'' being a synonym of ''concept'' is suggested by the definition of ''concept'' in Meriam-Webster 1913: &quot;An abstract general conception; a notion; a universal.&quot;&lt;ref&gt;http://www.websters1913.com/words/Concept&lt;/ref&gt;

One possible terminological deliberation and choice could be this. Even if ''notion'' is sometimes used synonymously with ''concept'', it is ''concept'' that is the headword of Britannica 1911 article, modern Britannica online and Stanford Encyclopedia of Philosophy; and thus, ''concept'' would be the preferred name and ''notion'' the dispreferred name (that's an if).

To support the above terminological choice, we can consider the rates in Google Ngram Viewer (GNV) of ''narrower concept'' vs. ''narrower notion''&lt;ref&gt;https://books.google.com/ngrams/graph?content=narrower+notion%2C+narrower+concept&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt; and ''broader concept'' vs. ''broader notion''&lt;ref&gt;https://books.google.com/ngrams/graph?content=broader+notion%2Cbroader+concept&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;. In both comparisons, the variant with ''concept'' comfortably wins but the variant with ''notion'' sees significant use.

Let us look at the NGV rates of ''undefined concepts'' and ''undefined notions''.&lt;ref&gt;https://books.google.com/ngrams/graph?content=undefined+concepts%2Cundefined+notions&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;. The picture is remarkable: in 19th century, ''undefined concepts'' sees almost no use, unlike ''undefined notions''; in 20th century, ''undefined concepts'' takes a lead while ''undefined notions'' still sees wide use. But it is above all ''terms'' that are undefined.&lt;ref&gt;https://books.google.com/ngrams/graph?content=undefined+*_NOUN&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;

Further reading:
* [https://www.merriam-webster.com/dictionary/notion notion], merriam-webster.com

== Names in other languages ==
Let us have a look at the names of the concept of concept in other languages. These can be meaningfully groupped by language family.
* English ''concept'', Italian ''concetto'', French ''concept''
* German ''Begriff'' but also ''Konzept'', Danish ''begreb''
* Czech ''pojem'', Russian понятие, Polish ''pojęcie''

== Frege's Begriffsschrift ==
Frege's Begriffsschrift can be translated as concept/conceptual script (collection of letters). From what I understand, in the formalism developed in Begriffsschrift, there is a one-to-one correspondence between concept letters and concepts; there remains no ambiguity.

Further reading:
* {{W|Begriffsschrift}}, wikipedia.org

==See also==
* [[Operationalization]]
* [[Concept clarification]]

==References==
&lt;references/&gt;

==Further reading==
* {{W|Concept}}, wikipedia.org
* [[S:1911 Encyclopædia Britannica/Concept]], wikisource.org
* [https://philpapers.org/browse/Concepts Concept] at PhilPapers
* &quot;[https://plato.stanford.edu/entries/concepts/ Concepts]&quot;. Stanford Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/concepts/ Concept]&quot;. Internet Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/theory-theory-of-concepts/ Theory–Theory of Concepts]&quot;. Internet Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/classical-theory-of-concepts/ Classical Theory of Concepts]&quot;. Internet Encyclopedia of Philosophy.
* [https://www.britannica.com/topic/concept concept], britannica.com
* [https://dictionary.apa.org/concept concept] in APA Dictionary of Psychology
* [https://dictionary.apa.org/disjunctive-concept disjunctive concept] in APA Dictionary of Psychology
* [https://dictionary.apa.org/conjunctive-concept conjunctive concept] in APA Dictionary of Psychology
* [https://www.semanticarts.com/the-importance-of-distinguishing-between-terms-and-concepts/ The Importance of Distinguishing between Terms and Concepts] by Michael Uschold, 2013
* [https://www.researchgate.net/publication/330146725_Concept_vs_Notion_and_Lexical_Meaning_What_is_the_Difference Concept vs Notion and Lexical Meaning: What is the Difference?] by Dinara Khairullina, 2018
* [https://www.margolisphilosophy.com/uploads/1/1/0/7/11073530/learningmatters.pdf Learning Matters: The Role of Learning in Concept Acquisition] by Eric Margolis and Stephen Laurence, margolisphilosophy.com
* [https://www.st-andrews.ac.uk/~slr/Medieval_signification.pdf Concepts and Meaning in Medieval Philosophy] by Stephen Read, st-andrews.ac.uk
* [http://www.gap5.de/proceedings/pdf/419-434_newen.pdf Die ungeklärte Natur der Begriffe Eine Analyse der ontologischen Diskussion] by Albert Newen, gap5.de

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This article by Dan Polansky is about the concept of concept. It gives a first idea, raises some questions, and collects interesting further reading publicly available online. (The article may later be moved to &quot;One man's look at concept&quot; since it occupies the Concept head that others could also want to use.)

==Introduction==

A concept is to be distinguished from a word or phrase. As a first approximation, a concept is a word meaning. Thus, a single word covers multiple concepts, and a single concept is named by multiple words. By means of example, the concept of cat-domestical-animal is named by word &quot;cat&quot; but also by word &quot;grimalkin&quot;. And the other way around, the word &quot;cat&quot; covers the concept of cat-domestic-animal but also cat-felid-feline. We use the hyphenated names to unambiguously identify the concepts, hoping that listing a set of synonyms will serve to disambigute. Thus, each WordNet's synset is intended to match a single concept.

However, there are multiple reasons to think a concept does not have to be a word meaning. One reason is that at least some concepts seem to exist without words. Thus, pre-linguistic humans were probably able to rank observed individuals, e.g. individual trees, under a single mental category. The same is probably true of other primates. Such a sophisticated animal as a chimpanzee cannot operate in their environment without this kind of conceptual ability, the ability to rank individual observed objects into classes, and to interconnect classes by meaningful relationships. Thus, concepts would be some kind of mental objects different from word meaning. It would still be true that words point to concepts, but not each concept would need to be a word meaning.

Sum of parts concepts are also concepts. Thus, white-cat is also a concept, even if it is just cat that is white. Thus, it would seem any noun phrase ambiguously identifies a concept. Indeed, noun-phrase-that-ambiguously-identified-a-concept is also a concept.

As was pointed above, there is no unique way to name concepts. Thus, we can say cat-domestical-animal or cat-animal-in-the-narrow-sense. The former seems much better.

==Concepts for individual entities==

Concepts may also correspond to individual instances, at least to abstract instances. Since, &quot;blueness&quot; is a name of concept, a quality, and that is an individual instance.

One may be less ready to consider concepts for concrete individual entities, including individual people or trees. It is unclear what qualities a concept for, say, Albert Einstein contains: it rather seems that the Kripkean view is of import, that &quot;Albert Einstein&quot; is a rigid designator rather than a package of properties to select by. If one accepts the Russelian theory of names as descriptions, there are also concepts for individual instances, having a certain selection of properties as uniquely identifying these instances. One may object that these concepts are not really belonging to the individuals since they would not belong to them across possible worlds. This needs clarification and discussion.

The idea of concepts corresponding to concrete individual entities via descriptions suggests that different descriptions uniquely identifying an individual entity in this world could correspond to difference concepts, Fregean senses. Thus, Hesperus would pick the individual entity via a different concept than Phosphorus: one has the appearance in the morning sky as part of the concept whereas the other one in the evening sky.

We may wonder whether the idea of Kripkean rigid designation can inspire us to construct a concept that applies across possible worlds without the selecting description so applying:
* The person, in this world or a possible world, who is in this world named Albert Einstein and who in this world has discovered special theory of relativity, regardless whether he did so in the particular world under consideration.
Here, the description would be used to pick the individual entity in this world, and the connection to possible worlds would be via the idea of trans-world numerical identity that disregards which parts of the selecting description are necessary and which merely contingent.

==Disjunctive concepts==

Literature also talks about concepts that are disjunctive, using OR to connect criteria. This raises questions about whether any OR combination is a concept, including the following:
* Multi-genus: A star or a house.
* Multi-genus: A star or a house or a mineral or an inflected form or a computable function.
* A person who is tall or is Albert Einstein or has ever been to Canada.

==Complexity of concepts==
Possible complexity of concepts points to one set of questions that are to be answered. For a start, does each of the following descriptions correspond to a concept?
* A cat that is white, weighs at least 10 kg and has not broken a vase.
* The concept of metric space in mathematics, that is, a structure that meets all of certain axioms.
Is there any limit to a complexity of description or a specification of a concept? To put it in technical terms, given a particular language of first order logic, does any formula with one free variable correspond to a concept? If it does, it seems impossible for each concept to fit into human consciousness.

==Phenomenal concepts==
A concept can probably be expressed in terms of the manner in which it is observed without knowledge of or differentiation of the underlying kind of entity. Thus, we would have the concept of a star as that which reveals itself as bright dot in the sky, without knowing whether the entity causing the dot is a ball of matter or a hole through which something shines or whether we are talking about a single kind of underlying entity rather than a single produced kind of observation. We can classify such stars into fixed stars, slowly moving stars and falling stars. Without deeper knowledge, we could think that some bright dots in the sky are a result of shining balls of matter whereas other bright dots are a results of fires shining through holes.

An example of a concept perhaps less phenomenal but still rather phenomenal is zebra, which can informally defined as striped horse-like animal. There are multiple species of zebra. What makes it phenomenal is that the selection of animals into the concept in part does not depend on underlying genetic affinities only but rather on external appearance. The underlying genetic affinities can be thought of as hidden essences, as if truer or deeper natural joints into which the natural world is cut, sliced or articulated.

==Name-based concepts==
A concept can be probably constructed based on words referring to things only. Thus, the following would be concepts:
* Any person named Peter.
* Any entity called cat in whatever sense of the word. (Generally multiple disparate genera.)
* Any entity called by the German word &quot;Katze&quot; in any sense of the word. (Allows embedding of foreign language.)
* Any entity described by the German phrase &quot;schwarze Katze&quot; in any senses of the component words. (Embedding of whole phrases.)
The above may seems strange, but it is often that which is cognitively given in a situation. Thus, if one hears unknown people talk in another room without seeing them, and if one hears &quot;Peter, come here please&quot;, a valid even if uncertain inference is that there is someone named Peter, which is not a proper noun manner of reference but rather a common noun manner and thus much more of a candidate to be or correspond to a concept. And &quot;someone named Peter&quot; is as much as one can validly think about that individual entity represented in some way in the mind, together with the inferred &quot;human&quot; and &quot;male&quot;. Peter has not spoken so there is not even a way to associate particular voice with that entity.

==Very specific concepts==
The possibility of great specificity of concepts needs to be analyzed. Thus, it is to be clarified whether the following are concepts:
* A person shown on the photograph so and so.
* A person with fingerprint so and so.
This question, combined with the one about complexity, leads to the question whether the following defines a concept:
* A person that corresponds to the knowledge that person X (who has seen a photograph) has about Albert Einstein.
The above would imply the existence of many detailed very specific concepts corresponding to individual entities, where different people would have a different concept about the individual entity based on what they know, and the concept would keep on being enlarged with more knowledge gained. There is something implausible about this idea. If one further combines the idea with the idea of arbitrarily combining concepts via OR, we could construct a concept corresponding to knowledge of, say, 50 individual people a certain person has. This construction seems to be at odds with the reason why the concept of concept was introduced in the first place, to correspond to something like relatively general categories under which observed things and other things fall, where &quot;relatively general&quot; is left unspecified.

==Equivalent definitions of the same class==
For individual entities, we have seen that Hesperus and Phosphorus could correspond to different concepts even if the same individual entity. There could be a similar situation with classes, for instance of ellipse. An ellipse can be defined as a set of points with a particular constant sum of distances from the focal points or as a set of points generated by applying sine and cosine function to a parameter. These two equivalent definitions identify the same class, but in a different manner, and if two different ways of accessing Hesperus correspond to different concepts, so could the two different ways of accessing the class of ellipses.

==Nativism of concepts==
One can ask whether some or all concepts are innate or rather learned. It seems implausible that complex concepts are innate. At the same time, it seems hard to believe that no concept at all is innate.

If we take natural kinds such as biological species or minerals as corresponding to concepts, it seems hard to believe they would be innate. There is not enough environmental information and adaptive survival/reproduction advantage for an organism to have them all innate. For instance, assuming that humans evolved in Africa, it is unclear how the evolutionary analogue of learning consisting in generic and phenotypic variation and elimination would learn about Americas-only species. Similar puzzle applies to learning about species living only deep in the ocean or the concept of bacteria.

By contrast, it is superficially plausible to think that e.g. the concepts of animal (excluding humans), tree and river could be innate, driven by applications relating to survival and reproduction, or more accurately genetic fitness, and based on environmental information readily available to humans no less than chimpanzees. 

The subject is covered in Stanford Encyclopedia of Philosophy.

==Coupling to language==
The opening paragraphs started with the first approximation that concepts are word meanings. This raises the questions:
* Are there any concepts without words?
* Does language enhance one's work with concepts?
* Does it cognitively matter whether a concept is bound to a word, even if a compound word, or rather a phrase?
We have answered the first question in the affirmative. The answer to the second question seems true as well. Thus, for a mathematical example, the concept of metric space is easier to work with and arrive at if the concept has a name (&quot;metric space&quot;) and if it is linked to other concept's names via axioms. Moreover, the use of new words and phrases forces one to create a classification or conceptual scheme that ranks individual objects under the word or phrase, forcing concept formation.

An interesting window on the concept-language coupling is offered by German (and similar languages), a language with a tendency to form long compounds. Does the existence of ready-made long German compounds enhance one's capacity to think and work with concepts?

The subject is covered in Stanford Encyclopedia of Philosophy.

Further reading:
* [https://plato.stanford.edu/entries/concepts/#ConNatLan 4. Concepts and natural language] in Concepts, Stanford Encyclopedia of Philosophy

==Definitions of concepts==
In so far as concepts are or correspond to word and phrase senses, they can be defined. One can define a concept using words, but that is ambiguous. One can less ambiguously define concepts using concepts, provided one finds a way to unambiguously identify concepts using words or their groups. Thus, one can define cat-domestic-animal as domestic-animal that meows. Here we use a hyphen convention reminiscent of the programming language LISP to identify concepts less ambiguously than single words can.

==Adjectival and verbal concepts==
Further to be clarified is whether there are adjectival concepts and verbal concepts; there are such word senses. Since, instead of saying X is blue, we may say X has the quality or state of blueness, and thus we do not need any concept for adjective &quot;blue&quot;. Similarly, instead of saying that X operates, we may say that X is in the process of operation. If concepts are by definition nominal (corresponding to nouns and noun phrases), here would be another difference between concept and word sense.

==Applications==
The concept of concept can be used to study human and animal behavior. The term &quot;concept&quot; is used in standards for thesauri for information retrieval and in a related interchange data format.

==Concept vs. notion==
It seems that some literature has some uses of the word ''notion'' that are synonymous with ''concept''. Uses of the form &quot;the notion of X&quot; where X is a noun or a noun phrase are suggestive of this kind of use. Narrower uses suggestive of synonymy are &quot;define the notion of X&quot;. An example of this apparent use is the publication ''What is a Word? About the Notion of Word'' from GRIN Verlag, apparently written by German native speaker or speakers.

One way to shed some light on this question is to use a German-English dictionary to translate the German word ''Begriff''. If we use Cambridge dictionary, we get two different translations as different senses, one notion and one concept.&lt;ref&gt;https://dictionary.cambridge.org/dictionary/german-english/begriff?q=Begriff&lt;/ref&gt;. Langenscheidt features both notion and concept as possible translations.&lt;ref&gt;https://en.langenscheidt.com/german-english/begriff&lt;/ref&gt; Collins mentions concept but not notion.&lt;ref&gt;https://www.collinsdictionary.com/dictionary/german-english/begriff&lt;/ref&gt;

A similar way is to start with Czech ''pojem''. Lingea gives both concept and notion as possible translations, as different senses.&lt;ref&gt;https://slovniky.lingea.cz/anglicko-cesky/pojem&lt;/ref&gt;

The above exercise could be expanded with coverage of other languages. Starting with the two languages picked and the results, it would seem possible that native English speakers do not usually use the word ''notion'' as a synonym of ''concept'' whereas non-native speakers could do so under influence of translation dictionaries, by picking the wrong translation for the sought sense. But this is just a guess.

''Notion'' being a synonym of ''concept'' is suggested by the definition of ''concept'' in Meriam-Webster 1913: &quot;An abstract general conception; a notion; a universal.&quot;&lt;ref&gt;http://www.websters1913.com/words/Concept&lt;/ref&gt;

One possible terminological deliberation and choice could be this. Even if ''notion'' is sometimes used synonymously with ''concept'', it is ''concept'' that is the headword of Britannica 1911 article, modern Britannica online and Stanford Encyclopedia of Philosophy; and thus, ''concept'' would be the preferred name and ''notion'' the dispreferred name (that's an if).

To support the above terminological choice, we can consider the rates in Google Ngram Viewer (GNV) of ''narrower concept'' vs. ''narrower notion''&lt;ref&gt;https://books.google.com/ngrams/graph?content=narrower+notion%2C+narrower+concept&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt; and ''broader concept'' vs. ''broader notion''&lt;ref&gt;https://books.google.com/ngrams/graph?content=broader+notion%2Cbroader+concept&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;. In both comparisons, the variant with ''concept'' comfortably wins but the variant with ''notion'' sees significant use.

Let us look at the NGV rates of ''undefined concepts'' and ''undefined notions''.&lt;ref&gt;https://books.google.com/ngrams/graph?content=undefined+concepts%2Cundefined+notions&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;. The picture is remarkable: in 19th century, ''undefined concepts'' sees almost no use, unlike ''undefined notions''; in 20th century, ''undefined concepts'' takes a lead while ''undefined notions'' still sees wide use. But it is above all ''terms'' that are undefined.&lt;ref&gt;https://books.google.com/ngrams/graph?content=undefined+*_NOUN&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3&lt;/ref&gt;

Further reading:
* [https://www.merriam-webster.com/dictionary/notion notion], merriam-webster.com

== Names in other languages ==
Let us have a look at the names of the concept of concept in other languages. These can be meaningfully groupped by language family.
* English ''concept'', Italian ''concetto'', French ''concept''
* German ''Begriff'' but also ''Konzept'', Danish ''begreb''
* Czech ''pojem'', Russian понятие, Polish ''pojęcie''

== Frege's Begriffsschrift ==
Frege's Begriffsschrift can be translated as concept/conceptual script (collection of letters). From what I understand, in the formalism developed in Begriffsschrift, there is a one-to-one correspondence between concept letters and concepts; there remains no ambiguity.

Further reading:
* {{W|Begriffsschrift}}, wikipedia.org

==See also==
* [[Operationalization]]
* [[Concept clarification]]

==References==
&lt;references/&gt;

==Further reading==
* {{W|Concept}}, wikipedia.org
* [[S:1911 Encyclopædia Britannica/Concept]], wikisource.org
* [https://philpapers.org/browse/Concepts Concept] at PhilPapers
* &quot;[https://plato.stanford.edu/entries/concepts/ Concepts]&quot;. Stanford Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/concepts/ Concept]&quot;. Internet Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/theory-theory-of-concepts/ Theory–Theory of Concepts]&quot;. Internet Encyclopedia of Philosophy.
* &quot;[https://iep.utm.edu/classical-theory-of-concepts/ Classical Theory of Concepts]&quot;. Internet Encyclopedia of Philosophy.
* [https://www.britannica.com/topic/concept concept], britannica.com
* [https://dictionary.apa.org/concept concept] in APA Dictionary of Psychology
* [https://dictionary.apa.org/disjunctive-concept disjunctive concept] in APA Dictionary of Psychology
* [https://dictionary.apa.org/conjunctive-concept conjunctive concept] in APA Dictionary of Psychology
* [https://www.semanticarts.com/the-importance-of-distinguishing-between-terms-and-concepts/ The Importance of Distinguishing between Terms and Concepts] by Michael Uschold, 2013
* [https://www.researchgate.net/publication/330146725_Concept_vs_Notion_and_Lexical_Meaning_What_is_the_Difference Concept vs Notion and Lexical Meaning: What is the Difference?] by Dinara Khairullina, 2018
* [https://www.margolisphilosophy.com/uploads/1/1/0/7/11073530/learningmatters.pdf Learning Matters: The Role of Learning in Concept Acquisition] by Eric Margolis and Stephen Laurence, margolisphilosophy.com
* [https://www.st-andrews.ac.uk/~slr/Medieval_signification.pdf Concepts and Meaning in Medieval Philosophy] by Stephen Read, st-andrews.ac.uk
* [http://www.gap5.de/proceedings/pdf/419-434_newen.pdf Die ungeklärte Natur der Begriffe Eine Analyse der ontologischen Diskussion] by Albert Newen, gap5.de

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    <title>African Arthropods/Chalcidoidea</title>
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The megadiverse Superfamily Chalcidoidea has more than 22 000 described species (&gt;2 800 in Africa) and there may be as many as 500 000 undescribed species. Most of these are tiny (less than 3 mm long), and are parasites of a wide variety of insects, spiders and mites, but some chalcidoid larvae eat plant material (in galls or seeds).&lt;ref name=Noyes2003&gt;Noyes, J.S. and Pitkin, B.R. 2003. Universal Chalcidoidea Database: About chalcidoids. The Natural History Museum, London. [https://www.nhm.ac.uk/our-science/data/chalcidoids/introduction.html]&lt;/ref&gt;&lt;ref name=vanNoort2023&gt;van Noort, Simon (2023) WaspWeb: Hymenoptera of the Afrotropical region. www.waspweb.org, accessed on 9 May 2023.&lt;/ref&gt;

[[File:South African chalcidoid wasps in 9 new families 2023.jpg|thumb|600px|South African Chalcidoid wasps moved to new (2018-2022) families: ''Metapelmus'' sp. from Eupelmidae to Metapelmatidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Bohpa maculata'' from Pteromalidae to Ceidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Spalangia'' sp. from Pteromalidae to Spalangiidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Spathopus'' sp. from Pteromalidae to Pirenidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Aperilampus'' sp. from Perilampidae to Chrysolampidae
&lt;ref name=Zhang2022&gt;Zhang, J., Heraty, J. M., Darling, C., Kresslein, R. L., Baker, A. J., Torréns, J., Rasplus, J. Y., Lemmon A. &amp; Moriarty Lemmon, E. (2022). Anchored phylogenomics and a revised classification of the planidial larva clade of jewel wasps (Hymenoptera: Chalcidoidea). Systematic Entomology, 47(2), 329-353.  [https://doi.org/10.1111/syen.12533 DOI]&lt;/ref&gt;; ''Megastigmus transvaalensis'' from Torymidae to Megastigmidae&lt;ref name=Janšta2018&gt;Janšta, P., Cruaud, A., Delvare, G., Genson, G., Heraty, J., Křížková, B. and Rasplus, J.Y. (2018). Torymidae (Hymenoptera, Chalcidoidea) revised: molecular phylogeny, circumscription and reclassification of the family with discussion of its biogeography and evolution of life‐history traits. Cladistics, 34(6), pp.627-651. [https://doi.org/10.1111/cla.12228 DOI]&lt;/ref&gt;; ''Lelaps'' or ''Dipara'' sp. from Pteromalidae to Diparidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Lachaisea brevimucro'' from Pteromalidae to Epichrysomallidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; and ''Solenura nigra'' from Pteromalidae to Lyciscidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;.]]

Before 2018, these species were placed in about 19 accepted families. However, over the past few decades, a host of studies investigating their biology, their morphology, and their DNA have suggested that many more families should be recognized. In 2022 a group of wasp taxonomists agreed to recognize about 50 families worldwide.&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt; 

Macro photography has allowed non-scientists a view into the world of these minute creatures, but learning more about them is a challenge. Even with high-quality macro photographs of free-flying wasps, expert hymenopterists are unable to identify most chalcidoids to species or genus level. Nevertheless, with the help of the specialists that contribute identifications on iNaturalist and [https://www.facebook.com/groups/HymenopteristsForum88/ Hymenopterists Forum], it has been possible to put together this page, illustrated with photographs of live wasps from 23 of the 35-odd chalcidoid families known in Africa; it is supplemented with photos of museum specimens from another four families.&lt;br&gt;
&lt;br&gt;
Many more images (mostly of museum specimens, including type specimens) can be found on [https://www.waspweb.org/Chalcidoidea/index.htm WaspWeb]. This includes photos of wasps from the following families not shown here: [https://www.waspweb.org/Chalcidoidea/Azotidae/index.htm Azotidae], [https://www.waspweb.org/Chalcidoidea/Cleonymidae/index.htm Cleonymidae], [https://www.waspweb.org/Chalcidoidea/Herbertiidae/index.htm Herbertiidae], [https://www.waspweb.org/Chalcidoidea/Heydeniidae/index.htm Heydeniidae], [https://www.waspweb.org/Chalcidoidea/Neanastatidae/index.htm Neanastatidae], and [https://www.waspweb.org/Chalcidoidea/Tanaostigmatidae/index.htm Tanaostigmatidae].&lt;br&gt;
&lt;br&gt;


===[[w:Agaonidae|Agaonidae]]===
&lt;gallery mode=packed heights=200&gt;
Agaonidae_2019_10_02_10_18_49_5304.jpg |''Ceratosolen galili'' female
Agaonidae 2019 10 10 10 17 37 3249b.jpg |''Elisabethiella comptoni'' female
Agaonidae 2019 10 10 10 46 58 3485.jpg |''Elisabethiella comptoni'' male
&lt;/gallery&gt;

===[[w:Aphelinidae|Aphelinidae]]===
&lt;gallery mode=packed heights=200&gt;
Aphelinidae 2022 05 19 3300.jpg|''[[w:Eretmocerus|Eretmocerus]]'' sp.
&lt;/gallery&gt;

===[[w:Calesidae|Calesidae]]===
&lt;gallery mode=packed heights=200&gt;
Cales 2023 07 01 iN 171704435.jpg|''Cales'' sp.; female
Cales 2023 07 01 iN 171706452.jpg|''Cales'' sp.; male
&lt;/gallery&gt;

===[[w:Ceidae|Ceidae]]===
&lt;gallery mode=packed heights=200&gt;
Bohpa 01 Mitroiu 2016.jpg|''Bohpa maculata'', female
Bopha 02 Mitroiu 2016.jpg|''Bohpa maculata'', female
&lt;/gallery&gt;

===[[w:Cerocephalidae|Cerocephalidae]]===
&lt;gallery mode=packed heights=200&gt;
Cerocephalidae 2023 06 05 iNat166277117 01.jpg
Cerocephalidae 2023 06 05 iNat166277117 02.jpg
&lt;/gallery&gt;

===[[:w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2019 10 09 17 26 38 2520.jpg|''Brachymeria'' sp.
Dirhinus 2019 07 28 7032.jpg|''Dirhinus'' sp.
Epitranus 2019 06 05 1831 Ced.jpg |''Epitranus'' sp.
Hockeria sp inat140473878 Jonathan Whitaker Oct 29, 2022.jpg

&lt;/gallery&gt;

===[[:w:Chrysolampidae|Chrysolampidae]]===
&lt;gallery mode=packed heights=200&gt;
Aperilampus sp.00.jpg|''Aperilampus'' sp., male
Aperilampus 2019 05 09 7697.jpg|''Aperilampus'' sp., female
&lt;/gallery&gt;

===[[:w:Diparidae|Diparidae]]===
&lt;gallery mode=packed heights=200&gt;
Conophorisca littoriticus inaturalist 356775210 2.jpg
Diparidae 2022 11 02 8329.jpg|Probably ''Lelaps'' sp. or ''Dipara'' sp. (male)
&lt;/gallery&gt;

===[[:w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Aenasius 2019 08 25 9691.jpg |''Aenasius'' sp.
Homalotylus 2019 08 16 8521.jpg|''Homalotylus'' sp.
&lt;/gallery&gt;

===[[:w:Eunotidae|Eunotidae]]===
&lt;gallery mode=packed heights=200&gt;
Scutellista iN 249602134 2024 10 02 11 57 50 4478.jpg|''Scutellista'' sp. at a ''Croton megalabotrys'' nectary
Scutellista iN 249602134.jpg|''Scutellista'' sp. on a ''Croton megalabotrys'' petiole
&lt;/gallery&gt;

===[[:w:Epichrysomallidae|Epichrysomallidae]]===
&lt;gallery mode=packed heights=200&gt;
Lachaisea brevimucro 2022 06 26 11 06 44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;

===[[:w:Eucharitidae|Eucharitidae]]===
&lt;gallery mode=packed heights=200&gt;
Hydrorhoa stevensoni iNat139715984 2022 10 12 5521.jpg|''Hydrorhoa stevensoni'', male
Stilbula 2020 01 02 9092.jpg |''Stilbula'' sp., female
Mateucharis glabra 2022 05 19 3435.jpg|''Mateucharis glabra'',  male
&lt;/gallery&gt;

===[[:w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Eulophinae 2019 06 10 16 16 23 1167.jpg |Subfamily Eulophinae (Unidentified)
Pediobius 2020 01 18 5090 PPot.jpg |''Pediobius'' sp. 
Tetrastichinae 2019 06 04 1502.jpg |Subfamily Tetrastichinae (Unidentified)
Neotrichoporoides 2019 08 20 8905.jpg |''Neotrichoporoides'' sp.
Neotrichoporoides 2019 06 28 4331.jpg |''Neotrichoporoides'' sp.
Eulophidae Inat 38422095 b.jpg|''Hemiptarsenus'' sp.
&lt;/gallery&gt;

===[[:w:Eupelmidae|Eupelmidae]]===
&lt;gallery mode=packed heights=200&gt;
Pentacladia 2019 08 04 7557.jpg |''Pentacladia'' sp.
Eupelmus 2019 08 16 8466.jpg |''Eupelmus'' sp.
Eupelmidae 2020 01 01 15 26 12 7925 G.jpg |''Eupelmidae'' sp.
&lt;/gallery&gt;

===[[:w:Eurytomidae|Eurytomidae]]===
&lt;gallery mode=packed heights=200&gt;
Eurytomidae00.jpg|Subfamily Eurytominae (Unidentified)
Sycophila 2019 07 06 0705.jpg |''Sycophila'' sp.
Sycophila 2019 08 24.jpg |''Sycophila'' sp.
Eurytominae 2019 10 08 17 12 49 0944Shi.jpg |Subfamily Eurytominae (Unidentified)
&lt;/gallery&gt;

===[[:w:Leucospidae|Leucospidae]]===
&lt;gallery mode=packed heights=200&gt;
Micrapion 2019 07 28 6711.jpg |''Micrapion'' sp.
Leucospis 2019 09 03 1195.jpg |''Leucospis'' sp.
&lt;/gallery&gt;

===[[:w:Lyciscidae|Lyciscidae]]===
&lt;gallery mode=packed heights=200&gt;
Solenura 2019 10 08 0781Shi.jpg|''Solenura nigra''
Solenura 2019 10 08 0786Shi.jpg|''Solenura nigra''
&lt;/gallery&gt;

===[[:w:Megastigmidae|Megastigmidae]]===
&lt;gallery mode=packed heights=200&gt;
File:Megastigmus.jpg|''Megastigmus transvaalensis''
&lt;/gallery&gt;

===[[:w:Metapelmatidae|Metapelmatidae]]===
&lt;gallery mode=packed heights=200&gt;
Metapelma 2019 12 24 3471 C.jpg |''Metapelma'' sp.
&lt;/gallery&gt;

===[[:w:Mymaridae|Mymaridae]]===
&lt;gallery mode=packed heights=200&gt;
Mymaridae 2021 12 12 10 44 20.jpg|''Polynema'' sp.
Mymaridae 2022 05 15 12 32 49.jpg|''Polynema'' sp. or ''Acmopolynema'' sp.
Polynema sagittaria (10.3897-zookeys.783.26872) Figure 1A.jpg |''Polynema sagittaria''
&lt;/gallery&gt;

===[[:w:Ormyridae|Ormyridae]]===
&lt;gallery mode=packed heights=200&gt;
Ormyrus males 2024 09 28 iN 245187710 04.jpg|Male ''Ormyrus'' wasps on a stem gall
Ormyrus iNat90764688.jpg|Female ''Ormyrus'' wasp ovipositing
&lt;/gallery&gt;

===[[:w:Perilampidae|Perilampidae]]===
&lt;gallery mode=packed heights=200&gt;
Perilampus 2019 07 09 15 00 54 1165.jpg|''Perilampus'' sp.
&lt;/gallery&gt;

===[[:w:Pirenidae|Pirenidae]]===
&lt;gallery mode=packed heights=200&gt;
File:Pirenidae 160886456 04.jpg|''Spathopus'' sp.
&lt;/gallery&gt;

===[[:w:Pteromalidae|Pteromalidae ]]===
&lt;gallery mode=packed heights=200&gt;
Halticoptera 2019 11 26 2173Cedara.jpg |''Halticoptera'' sp.
Philocaenus_rotundus_2019_10_10_3433.jpg|''Seres rotundus''
Otitesella 2019 08 25 32467865 9382.jpg|''Otitesella'' sp.
&lt;/gallery&gt;

===[[:w:Signiphoridae|Signiphoridae]]===
&lt;gallery mode=packed heights=200&gt;
Signiphora flavella Woolley &amp; Dal Molin 2017.jpg|''Signiphora flavella''
&lt;/gallery&gt;

===[[:w:Spalangiidae|Spalangiidae]]===
&lt;gallery mode=packed heights=200&gt;
Spalangia 2019 11 22 1217 Cedara.jpg |''Spalangia'' sp.
&lt;/gallery&gt;

===[[:w:Systasidae|Systasidae]]===
&lt;gallery mode=packed heights=200&gt;
Systasis 29313948-78.jpg|''Systasis'' sp.
Systasis 38824978-07.jpg|''Systasis'' sp.
&lt;/gallery&gt;

===[[:w:Tetracampidae|Tetracampidae]]===
&lt;gallery mode=packed heights=200&gt;
Afrocampe prinslooi Gumovsky 2018.jpg|''Afrocampe prinslooi''
&lt;/gallery&gt;

===[[:w:Torymidae|Torymidae ]]===
&lt;gallery mode=packed heights=200&gt;
Torymidae_2019_12_25_3502.jpg |Family Torymidae (unidentified)
Podagrion 2019 07 03 5701.jpg |''Podagrion'' sp.
Ecdamua 2019 12 30 7386 Cedara.jpg |''Ecdamua'' sp.
Pseudotorymus_2019_07_28_6571.jpg |''Pseudotorymus'' sp.
&lt;/gallery&gt;

===[[:w:Trichogrammatidae|Trichogrammatidae]]===
&lt;gallery mode=packed heights=200&gt;
Poropoea africana Laudonia 2017.jpg|''Poropoea africana''
&lt;/gallery&gt;

==References==
{{reflist}}

[[Category:African Arthropods]]</text>
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    <revision>
      <id>2810429</id>
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      <timestamp>2026-05-19T13:44:33Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
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      <comment>/* Diparidae */</comment>
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The megadiverse Superfamily Chalcidoidea has more than 22 000 described species (&gt;2 800 in Africa) and there may be as many as 500 000 undescribed species. Most of these are tiny (less than 3 mm long), and are parasites of a wide variety of insects, spiders and mites, but some chalcidoid larvae eat plant material (in galls or seeds).&lt;ref name=Noyes2003&gt;Noyes, J.S. and Pitkin, B.R. 2003. Universal Chalcidoidea Database: About chalcidoids. The Natural History Museum, London. [https://www.nhm.ac.uk/our-science/data/chalcidoids/introduction.html]&lt;/ref&gt;&lt;ref name=vanNoort2023&gt;van Noort, Simon (2023) WaspWeb: Hymenoptera of the Afrotropical region. www.waspweb.org, accessed on 9 May 2023.&lt;/ref&gt;

[[File:South African chalcidoid wasps in 9 new families 2023.jpg|thumb|600px|South African Chalcidoid wasps moved to new (2018-2022) families: ''Metapelmus'' sp. from Eupelmidae to Metapelmatidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Bohpa maculata'' from Pteromalidae to Ceidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Spalangia'' sp. from Pteromalidae to Spalangiidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Spathopus'' sp. from Pteromalidae to Pirenidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Aperilampus'' sp. from Perilampidae to Chrysolampidae
&lt;ref name=Zhang2022&gt;Zhang, J., Heraty, J. M., Darling, C., Kresslein, R. L., Baker, A. J., Torréns, J., Rasplus, J. Y., Lemmon A. &amp; Moriarty Lemmon, E. (2022). Anchored phylogenomics and a revised classification of the planidial larva clade of jewel wasps (Hymenoptera: Chalcidoidea). Systematic Entomology, 47(2), 329-353.  [https://doi.org/10.1111/syen.12533 DOI]&lt;/ref&gt;; ''Megastigmus transvaalensis'' from Torymidae to Megastigmidae&lt;ref name=Janšta2018&gt;Janšta, P., Cruaud, A., Delvare, G., Genson, G., Heraty, J., Křížková, B. and Rasplus, J.Y. (2018). Torymidae (Hymenoptera, Chalcidoidea) revised: molecular phylogeny, circumscription and reclassification of the family with discussion of its biogeography and evolution of life‐history traits. Cladistics, 34(6), pp.627-651. [https://doi.org/10.1111/cla.12228 DOI]&lt;/ref&gt;; ''Lelaps'' or ''Dipara'' sp. from Pteromalidae to Diparidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Lachaisea brevimucro'' from Pteromalidae to Epichrysomallidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; and ''Solenura nigra'' from Pteromalidae to Lyciscidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;.]]

Before 2018, these species were placed in about 19 accepted families. However, over the past few decades, a host of studies investigating their biology, their morphology, and their DNA have suggested that many more families should be recognized. In 2022 a group of wasp taxonomists agreed to recognize about 50 families worldwide.&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt; 

Macro photography has allowed non-scientists a view into the world of these minute creatures, but learning more about them is a challenge. Even with high-quality macro photographs of free-flying wasps, expert hymenopterists are unable to identify most chalcidoids to species or genus level. Nevertheless, with the help of the specialists that contribute identifications on iNaturalist and [https://www.facebook.com/groups/HymenopteristsForum88/ Hymenopterists Forum], it has been possible to put together this page, illustrated with photographs of live wasps from 23 of the 35-odd chalcidoid families known in Africa; it is supplemented with photos of museum specimens from another four families.&lt;br&gt;
&lt;br&gt;
Many more images (mostly of museum specimens, including type specimens) can be found on [https://www.waspweb.org/Chalcidoidea/index.htm WaspWeb]. This includes photos of wasps from the following families not shown here: [https://www.waspweb.org/Chalcidoidea/Azotidae/index.htm Azotidae], [https://www.waspweb.org/Chalcidoidea/Cleonymidae/index.htm Cleonymidae], [https://www.waspweb.org/Chalcidoidea/Herbertiidae/index.htm Herbertiidae], [https://www.waspweb.org/Chalcidoidea/Heydeniidae/index.htm Heydeniidae], [https://www.waspweb.org/Chalcidoidea/Neanastatidae/index.htm Neanastatidae], and [https://www.waspweb.org/Chalcidoidea/Tanaostigmatidae/index.htm Tanaostigmatidae].&lt;br&gt;
&lt;br&gt;


===[[w:Agaonidae|Agaonidae]]===
&lt;gallery mode=packed heights=200&gt;
Agaonidae_2019_10_02_10_18_49_5304.jpg |''Ceratosolen galili'' female
Agaonidae 2019 10 10 10 17 37 3249b.jpg |''Elisabethiella comptoni'' female
Agaonidae 2019 10 10 10 46 58 3485.jpg |''Elisabethiella comptoni'' male
&lt;/gallery&gt;

===[[w:Aphelinidae|Aphelinidae]]===
&lt;gallery mode=packed heights=200&gt;
Aphelinidae 2022 05 19 3300.jpg|''[[w:Eretmocerus|Eretmocerus]]'' sp.
&lt;/gallery&gt;

===[[w:Calesidae|Calesidae]]===
&lt;gallery mode=packed heights=200&gt;
Cales 2023 07 01 iN 171704435.jpg|''Cales'' sp.; female
Cales 2023 07 01 iN 171706452.jpg|''Cales'' sp.; male
&lt;/gallery&gt;

===[[w:Ceidae|Ceidae]]===
&lt;gallery mode=packed heights=200&gt;
Bohpa 01 Mitroiu 2016.jpg|''Bohpa maculata'', female
Bopha 02 Mitroiu 2016.jpg|''Bohpa maculata'', female
&lt;/gallery&gt;

===[[w:Cerocephalidae|Cerocephalidae]]===
&lt;gallery mode=packed heights=200&gt;
Cerocephalidae 2023 06 05 iNat166277117 01.jpg
Cerocephalidae 2023 06 05 iNat166277117 02.jpg
&lt;/gallery&gt;

===[[:w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2019 10 09 17 26 38 2520.jpg|''Brachymeria'' sp.
Dirhinus 2019 07 28 7032.jpg|''Dirhinus'' sp.
Epitranus 2019 06 05 1831 Ced.jpg |''Epitranus'' sp.
Hockeria sp inat140473878 Jonathan Whitaker Oct 29, 2022.jpg

&lt;/gallery&gt;

===[[:w:Chrysolampidae|Chrysolampidae]]===
&lt;gallery mode=packed heights=200&gt;
Aperilampus sp.00.jpg|''Aperilampus'' sp., male
Aperilampus 2019 05 09 7697.jpg|''Aperilampus'' sp., female
&lt;/gallery&gt;

===[[:w:Diparidae|Diparidae]]===
&lt;gallery mode=packed heights=200&gt;
Conophorisca littoriticus inaturalist 356775210 2.jpg|''Conophorisca littoriticus'' inaturalist 356775210 2.jpg
Diparidae 2022 11 02 8329.jpg|Probably ''Lelaps'' sp. or ''Dipara'' sp. (male)
&lt;/gallery&gt;

===[[:w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Aenasius 2019 08 25 9691.jpg |''Aenasius'' sp.
Homalotylus 2019 08 16 8521.jpg|''Homalotylus'' sp.
&lt;/gallery&gt;

===[[:w:Eunotidae|Eunotidae]]===
&lt;gallery mode=packed heights=200&gt;
Scutellista iN 249602134 2024 10 02 11 57 50 4478.jpg|''Scutellista'' sp. at a ''Croton megalabotrys'' nectary
Scutellista iN 249602134.jpg|''Scutellista'' sp. on a ''Croton megalabotrys'' petiole
&lt;/gallery&gt;

===[[:w:Epichrysomallidae|Epichrysomallidae]]===
&lt;gallery mode=packed heights=200&gt;
Lachaisea brevimucro 2022 06 26 11 06 44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;

===[[:w:Eucharitidae|Eucharitidae]]===
&lt;gallery mode=packed heights=200&gt;
Hydrorhoa stevensoni iNat139715984 2022 10 12 5521.jpg|''Hydrorhoa stevensoni'', male
Stilbula 2020 01 02 9092.jpg |''Stilbula'' sp., female
Mateucharis glabra 2022 05 19 3435.jpg|''Mateucharis glabra'',  male
&lt;/gallery&gt;

===[[:w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Eulophinae 2019 06 10 16 16 23 1167.jpg |Subfamily Eulophinae (Unidentified)
Pediobius 2020 01 18 5090 PPot.jpg |''Pediobius'' sp. 
Tetrastichinae 2019 06 04 1502.jpg |Subfamily Tetrastichinae (Unidentified)
Neotrichoporoides 2019 08 20 8905.jpg |''Neotrichoporoides'' sp.
Neotrichoporoides 2019 06 28 4331.jpg |''Neotrichoporoides'' sp.
Eulophidae Inat 38422095 b.jpg|''Hemiptarsenus'' sp.
&lt;/gallery&gt;

===[[:w:Eupelmidae|Eupelmidae]]===
&lt;gallery mode=packed heights=200&gt;
Pentacladia 2019 08 04 7557.jpg |''Pentacladia'' sp.
Eupelmus 2019 08 16 8466.jpg |''Eupelmus'' sp.
Eupelmidae 2020 01 01 15 26 12 7925 G.jpg |''Eupelmidae'' sp.
&lt;/gallery&gt;

===[[:w:Eurytomidae|Eurytomidae]]===
&lt;gallery mode=packed heights=200&gt;
Eurytomidae00.jpg|Subfamily Eurytominae (Unidentified)
Sycophila 2019 07 06 0705.jpg |''Sycophila'' sp.
Sycophila 2019 08 24.jpg |''Sycophila'' sp.
Eurytominae 2019 10 08 17 12 49 0944Shi.jpg |Subfamily Eurytominae (Unidentified)
&lt;/gallery&gt;

===[[:w:Leucospidae|Leucospidae]]===
&lt;gallery mode=packed heights=200&gt;
Micrapion 2019 07 28 6711.jpg |''Micrapion'' sp.
Leucospis 2019 09 03 1195.jpg |''Leucospis'' sp.
&lt;/gallery&gt;

===[[:w:Lyciscidae|Lyciscidae]]===
&lt;gallery mode=packed heights=200&gt;
Solenura 2019 10 08 0781Shi.jpg|''Solenura nigra''
Solenura 2019 10 08 0786Shi.jpg|''Solenura nigra''
&lt;/gallery&gt;

===[[:w:Megastigmidae|Megastigmidae]]===
&lt;gallery mode=packed heights=200&gt;
File:Megastigmus.jpg|''Megastigmus transvaalensis''
&lt;/gallery&gt;

===[[:w:Metapelmatidae|Metapelmatidae]]===
&lt;gallery mode=packed heights=200&gt;
Metapelma 2019 12 24 3471 C.jpg |''Metapelma'' sp.
&lt;/gallery&gt;

===[[:w:Mymaridae|Mymaridae]]===
&lt;gallery mode=packed heights=200&gt;
Mymaridae 2021 12 12 10 44 20.jpg|''Polynema'' sp.
Mymaridae 2022 05 15 12 32 49.jpg|''Polynema'' sp. or ''Acmopolynema'' sp.
Polynema sagittaria (10.3897-zookeys.783.26872) Figure 1A.jpg |''Polynema sagittaria''
&lt;/gallery&gt;

===[[:w:Ormyridae|Ormyridae]]===
&lt;gallery mode=packed heights=200&gt;
Ormyrus males 2024 09 28 iN 245187710 04.jpg|Male ''Ormyrus'' wasps on a stem gall
Ormyrus iNat90764688.jpg|Female ''Ormyrus'' wasp ovipositing
&lt;/gallery&gt;

===[[:w:Perilampidae|Perilampidae]]===
&lt;gallery mode=packed heights=200&gt;
Perilampus 2019 07 09 15 00 54 1165.jpg|''Perilampus'' sp.
&lt;/gallery&gt;

===[[:w:Pirenidae|Pirenidae]]===
&lt;gallery mode=packed heights=200&gt;
File:Pirenidae 160886456 04.jpg|''Spathopus'' sp.
&lt;/gallery&gt;

===[[:w:Pteromalidae|Pteromalidae ]]===
&lt;gallery mode=packed heights=200&gt;
Halticoptera 2019 11 26 2173Cedara.jpg |''Halticoptera'' sp.
Philocaenus_rotundus_2019_10_10_3433.jpg|''Seres rotundus''
Otitesella 2019 08 25 32467865 9382.jpg|''Otitesella'' sp.
&lt;/gallery&gt;

===[[:w:Signiphoridae|Signiphoridae]]===
&lt;gallery mode=packed heights=200&gt;
Signiphora flavella Woolley &amp; Dal Molin 2017.jpg|''Signiphora flavella''
&lt;/gallery&gt;

===[[:w:Spalangiidae|Spalangiidae]]===
&lt;gallery mode=packed heights=200&gt;
Spalangia 2019 11 22 1217 Cedara.jpg |''Spalangia'' sp.
&lt;/gallery&gt;

===[[:w:Systasidae|Systasidae]]===
&lt;gallery mode=packed heights=200&gt;
Systasis 29313948-78.jpg|''Systasis'' sp.
Systasis 38824978-07.jpg|''Systasis'' sp.
&lt;/gallery&gt;

===[[:w:Tetracampidae|Tetracampidae]]===
&lt;gallery mode=packed heights=200&gt;
Afrocampe prinslooi Gumovsky 2018.jpg|''Afrocampe prinslooi''
&lt;/gallery&gt;

===[[:w:Torymidae|Torymidae ]]===
&lt;gallery mode=packed heights=200&gt;
Torymidae_2019_12_25_3502.jpg |Family Torymidae (unidentified)
Podagrion 2019 07 03 5701.jpg |''Podagrion'' sp.
Ecdamua 2019 12 30 7386 Cedara.jpg |''Ecdamua'' sp.
Pseudotorymus_2019_07_28_6571.jpg |''Pseudotorymus'' sp.
&lt;/gallery&gt;

===[[:w:Trichogrammatidae|Trichogrammatidae]]===
&lt;gallery mode=packed heights=200&gt;
Poropoea africana Laudonia 2017.jpg|''Poropoea africana''
&lt;/gallery&gt;

==References==
{{reflist}}

[[Category:African Arthropods]]</text>
      <sha1>aecf1gf0vw7gxvuncynwhnthf21xno9</sha1>
    </revision>
    <revision>
      <id>2810430</id>
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      <timestamp>2026-05-19T13:44:56Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
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The megadiverse Superfamily Chalcidoidea has more than 22 000 described species (&gt;2 800 in Africa) and there may be as many as 500 000 undescribed species. Most of these are tiny (less than 3 mm long), and are parasites of a wide variety of insects, spiders and mites, but some chalcidoid larvae eat plant material (in galls or seeds).&lt;ref name=Noyes2003&gt;Noyes, J.S. and Pitkin, B.R. 2003. Universal Chalcidoidea Database: About chalcidoids. The Natural History Museum, London. [https://www.nhm.ac.uk/our-science/data/chalcidoids/introduction.html]&lt;/ref&gt;&lt;ref name=vanNoort2023&gt;van Noort, Simon (2023) WaspWeb: Hymenoptera of the Afrotropical region. www.waspweb.org, accessed on 9 May 2023.&lt;/ref&gt;

[[File:South African chalcidoid wasps in 9 new families 2023.jpg|thumb|600px|South African Chalcidoid wasps moved to new (2018-2022) families: ''Metapelmus'' sp. from Eupelmidae to Metapelmatidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Bohpa maculata'' from Pteromalidae to Ceidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Spalangia'' sp. from Pteromalidae to Spalangiidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Spathopus'' sp. from Pteromalidae to Pirenidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Aperilampus'' sp. from Perilampidae to Chrysolampidae
&lt;ref name=Zhang2022&gt;Zhang, J., Heraty, J. M., Darling, C., Kresslein, R. L., Baker, A. J., Torréns, J., Rasplus, J. Y., Lemmon A. &amp; Moriarty Lemmon, E. (2022). Anchored phylogenomics and a revised classification of the planidial larva clade of jewel wasps (Hymenoptera: Chalcidoidea). Systematic Entomology, 47(2), 329-353.  [https://doi.org/10.1111/syen.12533 DOI]&lt;/ref&gt;; ''Megastigmus transvaalensis'' from Torymidae to Megastigmidae&lt;ref name=Janšta2018&gt;Janšta, P., Cruaud, A., Delvare, G., Genson, G., Heraty, J., Křížková, B. and Rasplus, J.Y. (2018). Torymidae (Hymenoptera, Chalcidoidea) revised: molecular phylogeny, circumscription and reclassification of the family with discussion of its biogeography and evolution of life‐history traits. Cladistics, 34(6), pp.627-651. [https://doi.org/10.1111/cla.12228 DOI]&lt;/ref&gt;; ''Lelaps'' or ''Dipara'' sp. from Pteromalidae to Diparidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Lachaisea brevimucro'' from Pteromalidae to Epichrysomallidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; and ''Solenura nigra'' from Pteromalidae to Lyciscidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;.]]

Before 2018, these species were placed in about 19 accepted families. However, over the past few decades, a host of studies investigating their biology, their morphology, and their DNA have suggested that many more families should be recognized. In 2022 a group of wasp taxonomists agreed to recognize about 50 families worldwide.&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt; 

Macro photography has allowed non-scientists a view into the world of these minute creatures, but learning more about them is a challenge. Even with high-quality macro photographs of free-flying wasps, expert hymenopterists are unable to identify most chalcidoids to species or genus level. Nevertheless, with the help of the specialists that contribute identifications on iNaturalist and [https://www.facebook.com/groups/HymenopteristsForum88/ Hymenopterists Forum], it has been possible to put together this page, illustrated with photographs of live wasps from 23 of the 35-odd chalcidoid families known in Africa; it is supplemented with photos of museum specimens from another four families.&lt;br&gt;
&lt;br&gt;
Many more images (mostly of museum specimens, including type specimens) can be found on [https://www.waspweb.org/Chalcidoidea/index.htm WaspWeb]. This includes photos of wasps from the following families not shown here: [https://www.waspweb.org/Chalcidoidea/Azotidae/index.htm Azotidae], [https://www.waspweb.org/Chalcidoidea/Cleonymidae/index.htm Cleonymidae], [https://www.waspweb.org/Chalcidoidea/Herbertiidae/index.htm Herbertiidae], [https://www.waspweb.org/Chalcidoidea/Heydeniidae/index.htm Heydeniidae], [https://www.waspweb.org/Chalcidoidea/Neanastatidae/index.htm Neanastatidae], and [https://www.waspweb.org/Chalcidoidea/Tanaostigmatidae/index.htm Tanaostigmatidae].&lt;br&gt;
&lt;br&gt;


===[[w:Agaonidae|Agaonidae]]===
&lt;gallery mode=packed heights=200&gt;
Agaonidae_2019_10_02_10_18_49_5304.jpg |''Ceratosolen galili'' female
Agaonidae 2019 10 10 10 17 37 3249b.jpg |''Elisabethiella comptoni'' female
Agaonidae 2019 10 10 10 46 58 3485.jpg |''Elisabethiella comptoni'' male
&lt;/gallery&gt;

===[[w:Aphelinidae|Aphelinidae]]===
&lt;gallery mode=packed heights=200&gt;
Aphelinidae 2022 05 19 3300.jpg|''[[w:Eretmocerus|Eretmocerus]]'' sp.
&lt;/gallery&gt;

===[[w:Calesidae|Calesidae]]===
&lt;gallery mode=packed heights=200&gt;
Cales 2023 07 01 iN 171704435.jpg|''Cales'' sp.; female
Cales 2023 07 01 iN 171706452.jpg|''Cales'' sp.; male
&lt;/gallery&gt;

===[[w:Ceidae|Ceidae]]===
&lt;gallery mode=packed heights=200&gt;
Bohpa 01 Mitroiu 2016.jpg|''Bohpa maculata'', female
Bopha 02 Mitroiu 2016.jpg|''Bohpa maculata'', female
&lt;/gallery&gt;

===[[w:Cerocephalidae|Cerocephalidae]]===
&lt;gallery mode=packed heights=200&gt;
Cerocephalidae 2023 06 05 iNat166277117 01.jpg
Cerocephalidae 2023 06 05 iNat166277117 02.jpg
&lt;/gallery&gt;

===[[:w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2019 10 09 17 26 38 2520.jpg|''Brachymeria'' sp.
Dirhinus 2019 07 28 7032.jpg|''Dirhinus'' sp.
Epitranus 2019 06 05 1831 Ced.jpg |''Epitranus'' sp.
Hockeria sp inat140473878 Jonathan Whitaker Oct 29, 2022.jpg

&lt;/gallery&gt;

===[[:w:Chrysolampidae|Chrysolampidae]]===
&lt;gallery mode=packed heights=200&gt;
Aperilampus sp.00.jpg|''Aperilampus'' sp., male
Aperilampus 2019 05 09 7697.jpg|''Aperilampus'' sp., female
&lt;/gallery&gt;

===[[:w:Diparidae|Diparidae]]===
&lt;gallery mode=packed heights=200&gt;
Conophorisca littoriticus inaturalist 356775210 2.jpg|''Conophorisca littoriticus''
Diparidae 2022 11 02 8329.jpg|Probably ''Lelaps'' sp. or ''Dipara'' sp. (male)
&lt;/gallery&gt;

===[[:w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Aenasius 2019 08 25 9691.jpg |''Aenasius'' sp.
Homalotylus 2019 08 16 8521.jpg|''Homalotylus'' sp.
&lt;/gallery&gt;

===[[:w:Eunotidae|Eunotidae]]===
&lt;gallery mode=packed heights=200&gt;
Scutellista iN 249602134 2024 10 02 11 57 50 4478.jpg|''Scutellista'' sp. at a ''Croton megalabotrys'' nectary
Scutellista iN 249602134.jpg|''Scutellista'' sp. on a ''Croton megalabotrys'' petiole
&lt;/gallery&gt;

===[[:w:Epichrysomallidae|Epichrysomallidae]]===
&lt;gallery mode=packed heights=200&gt;
Lachaisea brevimucro 2022 06 26 11 06 44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;

===[[:w:Eucharitidae|Eucharitidae]]===
&lt;gallery mode=packed heights=200&gt;
Hydrorhoa stevensoni iNat139715984 2022 10 12 5521.jpg|''Hydrorhoa stevensoni'', male
Stilbula 2020 01 02 9092.jpg |''Stilbula'' sp., female
Mateucharis glabra 2022 05 19 3435.jpg|''Mateucharis glabra'',  male
&lt;/gallery&gt;

===[[:w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Eulophinae 2019 06 10 16 16 23 1167.jpg |Subfamily Eulophinae (Unidentified)
Pediobius 2020 01 18 5090 PPot.jpg |''Pediobius'' sp. 
Tetrastichinae 2019 06 04 1502.jpg |Subfamily Tetrastichinae (Unidentified)
Neotrichoporoides 2019 08 20 8905.jpg |''Neotrichoporoides'' sp.
Neotrichoporoides 2019 06 28 4331.jpg |''Neotrichoporoides'' sp.
Eulophidae Inat 38422095 b.jpg|''Hemiptarsenus'' sp.
&lt;/gallery&gt;

===[[:w:Eupelmidae|Eupelmidae]]===
&lt;gallery mode=packed heights=200&gt;
Pentacladia 2019 08 04 7557.jpg |''Pentacladia'' sp.
Eupelmus 2019 08 16 8466.jpg |''Eupelmus'' sp.
Eupelmidae 2020 01 01 15 26 12 7925 G.jpg |''Eupelmidae'' sp.
&lt;/gallery&gt;

===[[:w:Eurytomidae|Eurytomidae]]===
&lt;gallery mode=packed heights=200&gt;
Eurytomidae00.jpg|Subfamily Eurytominae (Unidentified)
Sycophila 2019 07 06 0705.jpg |''Sycophila'' sp.
Sycophila 2019 08 24.jpg |''Sycophila'' sp.
Eurytominae 2019 10 08 17 12 49 0944Shi.jpg |Subfamily Eurytominae (Unidentified)
&lt;/gallery&gt;

===[[:w:Leucospidae|Leucospidae]]===
&lt;gallery mode=packed heights=200&gt;
Micrapion 2019 07 28 6711.jpg |''Micrapion'' sp.
Leucospis 2019 09 03 1195.jpg |''Leucospis'' sp.
&lt;/gallery&gt;

===[[:w:Lyciscidae|Lyciscidae]]===
&lt;gallery mode=packed heights=200&gt;
Solenura 2019 10 08 0781Shi.jpg|''Solenura nigra''
Solenura 2019 10 08 0786Shi.jpg|''Solenura nigra''
&lt;/gallery&gt;

===[[:w:Megastigmidae|Megastigmidae]]===
&lt;gallery mode=packed heights=200&gt;
File:Megastigmus.jpg|''Megastigmus transvaalensis''
&lt;/gallery&gt;

===[[:w:Metapelmatidae|Metapelmatidae]]===
&lt;gallery mode=packed heights=200&gt;
Metapelma 2019 12 24 3471 C.jpg |''Metapelma'' sp.
&lt;/gallery&gt;

===[[:w:Mymaridae|Mymaridae]]===
&lt;gallery mode=packed heights=200&gt;
Mymaridae 2021 12 12 10 44 20.jpg|''Polynema'' sp.
Mymaridae 2022 05 15 12 32 49.jpg|''Polynema'' sp. or ''Acmopolynema'' sp.
Polynema sagittaria (10.3897-zookeys.783.26872) Figure 1A.jpg |''Polynema sagittaria''
&lt;/gallery&gt;

===[[:w:Ormyridae|Ormyridae]]===
&lt;gallery mode=packed heights=200&gt;
Ormyrus males 2024 09 28 iN 245187710 04.jpg|Male ''Ormyrus'' wasps on a stem gall
Ormyrus iNat90764688.jpg|Female ''Ormyrus'' wasp ovipositing
&lt;/gallery&gt;

===[[:w:Perilampidae|Perilampidae]]===
&lt;gallery mode=packed heights=200&gt;
Perilampus 2019 07 09 15 00 54 1165.jpg|''Perilampus'' sp.
&lt;/gallery&gt;

===[[:w:Pirenidae|Pirenidae]]===
&lt;gallery mode=packed heights=200&gt;
File:Pirenidae 160886456 04.jpg|''Spathopus'' sp.
&lt;/gallery&gt;

===[[:w:Pteromalidae|Pteromalidae ]]===
&lt;gallery mode=packed heights=200&gt;
Halticoptera 2019 11 26 2173Cedara.jpg |''Halticoptera'' sp.
Philocaenus_rotundus_2019_10_10_3433.jpg|''Seres rotundus''
Otitesella 2019 08 25 32467865 9382.jpg|''Otitesella'' sp.
&lt;/gallery&gt;

===[[:w:Signiphoridae|Signiphoridae]]===
&lt;gallery mode=packed heights=200&gt;
Signiphora flavella Woolley &amp; Dal Molin 2017.jpg|''Signiphora flavella''
&lt;/gallery&gt;

===[[:w:Spalangiidae|Spalangiidae]]===
&lt;gallery mode=packed heights=200&gt;
Spalangia 2019 11 22 1217 Cedara.jpg |''Spalangia'' sp.
&lt;/gallery&gt;

===[[:w:Systasidae|Systasidae]]===
&lt;gallery mode=packed heights=200&gt;
Systasis 29313948-78.jpg|''Systasis'' sp.
Systasis 38824978-07.jpg|''Systasis'' sp.
&lt;/gallery&gt;

===[[:w:Tetracampidae|Tetracampidae]]===
&lt;gallery mode=packed heights=200&gt;
Afrocampe prinslooi Gumovsky 2018.jpg|''Afrocampe prinslooi''
&lt;/gallery&gt;

===[[:w:Torymidae|Torymidae ]]===
&lt;gallery mode=packed heights=200&gt;
Torymidae_2019_12_25_3502.jpg |Family Torymidae (unidentified)
Podagrion 2019 07 03 5701.jpg |''Podagrion'' sp.
Ecdamua 2019 12 30 7386 Cedara.jpg |''Ecdamua'' sp.
Pseudotorymus_2019_07_28_6571.jpg |''Pseudotorymus'' sp.
&lt;/gallery&gt;

===[[:w:Trichogrammatidae|Trichogrammatidae]]===
&lt;gallery mode=packed heights=200&gt;
Poropoea africana Laudonia 2017.jpg|''Poropoea africana''
&lt;/gallery&gt;

==References==
{{reflist}}

[[Category:African Arthropods]]</text>
      <sha1>arlmxkqdh63kp5m1ache3t7gu1q25xh</sha1>
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    <revision>
      <id>2810578</id>
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        <username>Alandmanson</username>
        <id>1669821</id>
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      <comment>/* Eulophidae */</comment>
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The megadiverse Superfamily Chalcidoidea has more than 22 000 described species (&gt;2 800 in Africa) and there may be as many as 500 000 undescribed species. Most of these are tiny (less than 3 mm long), and are parasites of a wide variety of insects, spiders and mites, but some chalcidoid larvae eat plant material (in galls or seeds).&lt;ref name=Noyes2003&gt;Noyes, J.S. and Pitkin, B.R. 2003. Universal Chalcidoidea Database: About chalcidoids. The Natural History Museum, London. [https://www.nhm.ac.uk/our-science/data/chalcidoids/introduction.html]&lt;/ref&gt;&lt;ref name=vanNoort2023&gt;van Noort, Simon (2023) WaspWeb: Hymenoptera of the Afrotropical region. www.waspweb.org, accessed on 9 May 2023.&lt;/ref&gt;

[[File:South African chalcidoid wasps in 9 new families 2023.jpg|thumb|600px|South African Chalcidoid wasps moved to new (2018-2022) families: ''Metapelmus'' sp. from Eupelmidae to Metapelmatidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Bohpa maculata'' from Pteromalidae to Ceidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Spalangia'' sp. from Pteromalidae to Spalangiidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Spathopus'' sp. from Pteromalidae to Pirenidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Aperilampus'' sp. from Perilampidae to Chrysolampidae
&lt;ref name=Zhang2022&gt;Zhang, J., Heraty, J. M., Darling, C., Kresslein, R. L., Baker, A. J., Torréns, J., Rasplus, J. Y., Lemmon A. &amp; Moriarty Lemmon, E. (2022). Anchored phylogenomics and a revised classification of the planidial larva clade of jewel wasps (Hymenoptera: Chalcidoidea). Systematic Entomology, 47(2), 329-353.  [https://doi.org/10.1111/syen.12533 DOI]&lt;/ref&gt;; ''Megastigmus transvaalensis'' from Torymidae to Megastigmidae&lt;ref name=Janšta2018&gt;Janšta, P., Cruaud, A., Delvare, G., Genson, G., Heraty, J., Křížková, B. and Rasplus, J.Y. (2018). Torymidae (Hymenoptera, Chalcidoidea) revised: molecular phylogeny, circumscription and reclassification of the family with discussion of its biogeography and evolution of life‐history traits. Cladistics, 34(6), pp.627-651. [https://doi.org/10.1111/cla.12228 DOI]&lt;/ref&gt;; ''Lelaps'' or ''Dipara'' sp. from Pteromalidae to Diparidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Lachaisea brevimucro'' from Pteromalidae to Epichrysomallidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; and ''Solenura nigra'' from Pteromalidae to Lyciscidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;.]]

Before 2018, these species were placed in about 19 accepted families. However, over the past few decades, a host of studies investigating their biology, their morphology, and their DNA have suggested that many more families should be recognized. In 2022 a group of wasp taxonomists agreed to recognize about 50 families worldwide.&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt; 

Macro photography has allowed non-scientists a view into the world of these minute creatures, but learning more about them is a challenge. Even with high-quality macro photographs of free-flying wasps, expert hymenopterists are unable to identify most chalcidoids to species or genus level. Nevertheless, with the help of the specialists that contribute identifications on iNaturalist and [https://www.facebook.com/groups/HymenopteristsForum88/ Hymenopterists Forum], it has been possible to put together this page, illustrated with photographs of live wasps from 23 of the 35-odd chalcidoid families known in Africa; it is supplemented with photos of museum specimens from another four families.&lt;br&gt;
&lt;br&gt;
Many more images (mostly of museum specimens, including type specimens) can be found on [https://www.waspweb.org/Chalcidoidea/index.htm WaspWeb]. This includes photos of wasps from the following families not shown here: [https://www.waspweb.org/Chalcidoidea/Azotidae/index.htm Azotidae], [https://www.waspweb.org/Chalcidoidea/Cleonymidae/index.htm Cleonymidae], [https://www.waspweb.org/Chalcidoidea/Herbertiidae/index.htm Herbertiidae], [https://www.waspweb.org/Chalcidoidea/Heydeniidae/index.htm Heydeniidae], [https://www.waspweb.org/Chalcidoidea/Neanastatidae/index.htm Neanastatidae], and [https://www.waspweb.org/Chalcidoidea/Tanaostigmatidae/index.htm Tanaostigmatidae].&lt;br&gt;
&lt;br&gt;


===[[w:Agaonidae|Agaonidae]]===
&lt;gallery mode=packed heights=200&gt;
Agaonidae_2019_10_02_10_18_49_5304.jpg |''Ceratosolen galili'' female
Agaonidae 2019 10 10 10 17 37 3249b.jpg |''Elisabethiella comptoni'' female
Agaonidae 2019 10 10 10 46 58 3485.jpg |''Elisabethiella comptoni'' male
&lt;/gallery&gt;

===[[w:Aphelinidae|Aphelinidae]]===
&lt;gallery mode=packed heights=200&gt;
Aphelinidae 2022 05 19 3300.jpg|''[[w:Eretmocerus|Eretmocerus]]'' sp.
&lt;/gallery&gt;

===[[w:Calesidae|Calesidae]]===
&lt;gallery mode=packed heights=200&gt;
Cales 2023 07 01 iN 171704435.jpg|''Cales'' sp.; female
Cales 2023 07 01 iN 171706452.jpg|''Cales'' sp.; male
&lt;/gallery&gt;

===[[w:Ceidae|Ceidae]]===
&lt;gallery mode=packed heights=200&gt;
Bohpa 01 Mitroiu 2016.jpg|''Bohpa maculata'', female
Bopha 02 Mitroiu 2016.jpg|''Bohpa maculata'', female
&lt;/gallery&gt;

===[[w:Cerocephalidae|Cerocephalidae]]===
&lt;gallery mode=packed heights=200&gt;
Cerocephalidae 2023 06 05 iNat166277117 01.jpg
Cerocephalidae 2023 06 05 iNat166277117 02.jpg
&lt;/gallery&gt;

===[[:w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2019 10 09 17 26 38 2520.jpg|''Brachymeria'' sp.
Dirhinus 2019 07 28 7032.jpg|''Dirhinus'' sp.
Epitranus 2019 06 05 1831 Ced.jpg |''Epitranus'' sp.
Hockeria sp inat140473878 Jonathan Whitaker Oct 29, 2022.jpg

&lt;/gallery&gt;

===[[:w:Chrysolampidae|Chrysolampidae]]===
&lt;gallery mode=packed heights=200&gt;
Aperilampus sp.00.jpg|''Aperilampus'' sp., male
Aperilampus 2019 05 09 7697.jpg|''Aperilampus'' sp., female
&lt;/gallery&gt;

===[[:w:Diparidae|Diparidae]]===
&lt;gallery mode=packed heights=200&gt;
Conophorisca littoriticus inaturalist 356775210 2.jpg|''Conophorisca littoriticus''
Diparidae 2022 11 02 8329.jpg|Probably ''Lelaps'' sp. or ''Dipara'' sp. (male)
&lt;/gallery&gt;

===[[:w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Aenasius 2019 08 25 9691.jpg |''Aenasius'' sp.
Homalotylus 2019 08 16 8521.jpg|''Homalotylus'' sp.
&lt;/gallery&gt;

===[[:w:Eunotidae|Eunotidae]]===
&lt;gallery mode=packed heights=200&gt;
Scutellista iN 249602134 2024 10 02 11 57 50 4478.jpg|''Scutellista'' sp. at a ''Croton megalabotrys'' nectary
Scutellista iN 249602134.jpg|''Scutellista'' sp. on a ''Croton megalabotrys'' petiole
&lt;/gallery&gt;

===[[:w:Epichrysomallidae|Epichrysomallidae]]===
&lt;gallery mode=packed heights=200&gt;
Lachaisea brevimucro 2022 06 26 11 06 44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;

===[[:w:Eucharitidae|Eucharitidae]]===
&lt;gallery mode=packed heights=200&gt;
Hydrorhoa stevensoni iNat139715984 2022 10 12 5521.jpg|''Hydrorhoa stevensoni'', male
Stilbula 2020 01 02 9092.jpg |''Stilbula'' sp., female
Mateucharis glabra 2022 05 19 3435.jpg|''Mateucharis glabra'',  male
&lt;/gallery&gt;

===[[:w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Eulophinae 2019 06 10 16 16 23 1167.jpg |Subfamily Eulophinae (Unidentified)
Eulophidae 2025-09-21 inaturalist 315779038 01.jpg|''Euplectromorpha variegata''
Pediobius 2020 01 18 5090 PPot.jpg |''Pediobius'' sp. 
Tetrastichinae 2019 06 04 1502.jpg |Subfamily Tetrastichinae (Unidentified)
Neotrichoporoides 2019 08 20 8905.jpg |''Neotrichoporoides'' sp.
Neotrichoporoides 2019 06 28 4331.jpg |''Neotrichoporoides'' sp.
Eulophidae Inat 38422095 b.jpg|''Hemiptarsenus'' sp.
&lt;/gallery&gt;

===[[:w:Eupelmidae|Eupelmidae]]===
&lt;gallery mode=packed heights=200&gt;
Pentacladia 2019 08 04 7557.jpg |''Pentacladia'' sp.
Eupelmus 2019 08 16 8466.jpg |''Eupelmus'' sp.
Eupelmidae 2020 01 01 15 26 12 7925 G.jpg |''Eupelmidae'' sp.
&lt;/gallery&gt;

===[[:w:Eurytomidae|Eurytomidae]]===
&lt;gallery mode=packed heights=200&gt;
Eurytomidae00.jpg|Subfamily Eurytominae (Unidentified)
Sycophila 2019 07 06 0705.jpg |''Sycophila'' sp.
Sycophila 2019 08 24.jpg |''Sycophila'' sp.
Eurytominae 2019 10 08 17 12 49 0944Shi.jpg |Subfamily Eurytominae (Unidentified)
&lt;/gallery&gt;

===[[:w:Leucospidae|Leucospidae]]===
&lt;gallery mode=packed heights=200&gt;
Micrapion 2019 07 28 6711.jpg |''Micrapion'' sp.
Leucospis 2019 09 03 1195.jpg |''Leucospis'' sp.
&lt;/gallery&gt;

===[[:w:Lyciscidae|Lyciscidae]]===
&lt;gallery mode=packed heights=200&gt;
Solenura 2019 10 08 0781Shi.jpg|''Solenura nigra''
Solenura 2019 10 08 0786Shi.jpg|''Solenura nigra''
&lt;/gallery&gt;

===[[:w:Megastigmidae|Megastigmidae]]===
&lt;gallery mode=packed heights=200&gt;
File:Megastigmus.jpg|''Megastigmus transvaalensis''
&lt;/gallery&gt;

===[[:w:Metapelmatidae|Metapelmatidae]]===
&lt;gallery mode=packed heights=200&gt;
Metapelma 2019 12 24 3471 C.jpg |''Metapelma'' sp.
&lt;/gallery&gt;

===[[:w:Mymaridae|Mymaridae]]===
&lt;gallery mode=packed heights=200&gt;
Mymaridae 2021 12 12 10 44 20.jpg|''Polynema'' sp.
Mymaridae 2022 05 15 12 32 49.jpg|''Polynema'' sp. or ''Acmopolynema'' sp.
Polynema sagittaria (10.3897-zookeys.783.26872) Figure 1A.jpg |''Polynema sagittaria''
&lt;/gallery&gt;

===[[:w:Ormyridae|Ormyridae]]===
&lt;gallery mode=packed heights=200&gt;
Ormyrus males 2024 09 28 iN 245187710 04.jpg|Male ''Ormyrus'' wasps on a stem gall
Ormyrus iNat90764688.jpg|Female ''Ormyrus'' wasp ovipositing
&lt;/gallery&gt;

===[[:w:Perilampidae|Perilampidae]]===
&lt;gallery mode=packed heights=200&gt;
Perilampus 2019 07 09 15 00 54 1165.jpg|''Perilampus'' sp.
&lt;/gallery&gt;

===[[:w:Pirenidae|Pirenidae]]===
&lt;gallery mode=packed heights=200&gt;
File:Pirenidae 160886456 04.jpg|''Spathopus'' sp.
&lt;/gallery&gt;

===[[:w:Pteromalidae|Pteromalidae ]]===
&lt;gallery mode=packed heights=200&gt;
Halticoptera 2019 11 26 2173Cedara.jpg |''Halticoptera'' sp.
Philocaenus_rotundus_2019_10_10_3433.jpg|''Seres rotundus''
Otitesella 2019 08 25 32467865 9382.jpg|''Otitesella'' sp.
&lt;/gallery&gt;

===[[:w:Signiphoridae|Signiphoridae]]===
&lt;gallery mode=packed heights=200&gt;
Signiphora flavella Woolley &amp; Dal Molin 2017.jpg|''Signiphora flavella''
&lt;/gallery&gt;

===[[:w:Spalangiidae|Spalangiidae]]===
&lt;gallery mode=packed heights=200&gt;
Spalangia 2019 11 22 1217 Cedara.jpg |''Spalangia'' sp.
&lt;/gallery&gt;

===[[:w:Systasidae|Systasidae]]===
&lt;gallery mode=packed heights=200&gt;
Systasis 29313948-78.jpg|''Systasis'' sp.
Systasis 38824978-07.jpg|''Systasis'' sp.
&lt;/gallery&gt;

===[[:w:Tetracampidae|Tetracampidae]]===
&lt;gallery mode=packed heights=200&gt;
Afrocampe prinslooi Gumovsky 2018.jpg|''Afrocampe prinslooi''
&lt;/gallery&gt;

===[[:w:Torymidae|Torymidae ]]===
&lt;gallery mode=packed heights=200&gt;
Torymidae_2019_12_25_3502.jpg |Family Torymidae (unidentified)
Podagrion 2019 07 03 5701.jpg |''Podagrion'' sp.
Ecdamua 2019 12 30 7386 Cedara.jpg |''Ecdamua'' sp.
Pseudotorymus_2019_07_28_6571.jpg |''Pseudotorymus'' sp.
&lt;/gallery&gt;

===[[:w:Trichogrammatidae|Trichogrammatidae]]===
&lt;gallery mode=packed heights=200&gt;
Poropoea africana Laudonia 2017.jpg|''Poropoea africana''
&lt;/gallery&gt;

==References==
{{reflist}}

[[Category:African Arthropods]]</text>
      <sha1>f2zpbk2twrpg3hi64ys5jl2woiyolye</sha1>
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  </page>
  <page>
    <title>Corruption</title>
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      <timestamp>2026-05-19T16:09:10Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
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      <minor />
      <comment>project box(es)</comment>
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      <text bytes="10547" sha1="dsgximnaaolklqqunoutlm93hqbnhyw" xml:space="preserve">{{course}}
{{uncategorized}}
This comprehensive course aims to provide you with a deep understanding of corruption, its causes, effects, and strategies for combating it. Whether you're a student, researcher, or concerned citizen, this course will equip you with the knowledge and tools to tackle one of society's most pressing issues.

===Behavioural Objectives===
At the of this lesson learner's should be able to;

# Define Corruption
# Name types of corruption
# Identify the Causes Of corruption
# Understand the effects of corruption
# List the Strategies for combating corruption and how it can be prevented 

====Definition Of Corruption====
Corruption Simply means abuse of entrusted power for private gain.
Corruption involves dishonest actions by individuals or groups in power, such as accepting bribes, deceiving, or defrauding, often within organizations like governments or businesses.&lt;ref&gt;{{Cite web|url=https://www.transparency.org/en/what-is-corruption|title=What is corruption?|website=Transparency.org|language=en|access-date=2023-08-12}}&lt;/ref&gt;


====Types Of Corruption====
Corruption can be classified mainly into two
Petit Corruption and Grand Corruption 
#Petty corruption refers to situations involving relatively modest amounts of money, which may appear less impactful at a national level. Yet, when considered cumulatively, its effects can surpass those of grand corruption. It encompasses activities such as making unauthorized payments for medical appointments, obtaining positions in public schools, navigating checkpoints, advancing professional careers, and exerting influence over court decisions.
#Grand corruption is characterized by the involvement of substantial financial amounts, leading to significant and enduring repercussions for a nation. Examples include government officials misappropriating funds intended for public initiatives, inflating lawmakers' salaries to secure favorable legislation for specific individuals or groups, colluding with assessment authorities to grant environmentally harmful projects to extractive industries, assigning public contracts to unqualified bidders, and accepting subpar quality in the execution of public projects.&lt;ref&gt;{{Cite web|url=https://www.minpostel.gov.cm/index.php/fr/cellule-anti-corruption/296-types-and-impact-of-corruption|title=Types and Impact of Corruption|website=Ministry of Posts and Telecommunications, Cameroon|language=fr|access-date=2023-08-12}}&lt;/ref&gt;

====Causes Of Corruption====
The followings are the major Causes of corruption;
Greed of money, desires.
Higher levels of market and political monopolization.
Low levels of democracy, weak civil participation and low political transparency.
Higher levels of bureaucracy and inefficient administrative structures.
Low press freedom.&lt;ref&gt;{{Cite journal
|last=Šumah
|first=Štefan
|title=Corruption, Causes and Consequences
|journal=Open Access Peer-Reviewed Chapter
|date=February 21, 2018
|doi=10.5772/intechopen.72953
|url=https://www.intechopen.com/chapters/58969
|access-date=2023-08-12
}}&lt;/ref&gt;
#Greed and Desires: The pursuit of money and personal desires can drive individuals to engage in corrupt practices.
#Market and Political Monopolization: Higher levels of monopolization in markets and political spheres can foster an environment conducive to corruption.
#Democracy and Civil Participation:  Low levels of democracy, weak civil participation, and limited political transparency can create conditions where corruption thrives.
#Bureaucracy and Administrative Inefficiency: Higher levels of bureaucracy and inefficient administrative structures provide opportunities for corrupt behavior.
#Press Freedom: Low press freedom can limit the ability of media to expose and address corruption, allowing it to flourish in secrecy.

====Effects Of Corruption====


1. Economic Impact: Corruption adversely affects investment, both domestic and foreign. It hampers economic growth, reduces foreign direct investment and capital inflows, and distorts foreign trade and aid.

2. Inequality: Corruption contributes to inequality within societies. It diverts resources away from public services and benefits, leading to unequal distribution of opportunities and outcomes.

3. Government Services: Corrupt practices compromise the quality and efficiency of government services. Public expenditure and essential services like healthcare and education can be compromised due to diverted funds.

4. Growth Inhibition: Corruption stifles overall economic growth. It erodes trust in institutions and dampens entrepreneurial activities, leading to reduced innovation and development.

5. Shadow Economy and Crime: Corruption fosters a conducive environment for the growth of shadow economies and criminal activities. Illicit practices thrive when institutions are weakened by corruption.

6. Democracy and Political System: Corruption undermines the integrity of democratic processes. It erodes faith in the political system and can lead to manipulation of elections and distortion of representation.
&lt;ref&gt;{{Cite journal
|last=Šumah
|first=Štefan
|title=Corruption, Causes and Consequences
|journal=Open Access Peer-Reviewed Chapter
|date=February 21, 2018
|doi=10.5772/intechopen.72953
|url=https://www.intechopen.com/chapters/58969
|access-date=2023-08-12
}}&lt;/ref&gt;

====How Corruption Can be prevented====
The prevention of corruption hinges on the adoption of a comprehensive strategy that amalgamates a diverse array of measures. By integrating various methods, societies can construct a fortified framework aimed at proactively countering the influence of corruption. This holistic approach prioritizes principles such as transparency, accountability, and ethical conduct, collectively working towards creating an environment resistant to corrupt practices. Through these concerted efforts, the multifaceted approach strives to safeguard institutions, enhance governance, and ensure a fair and just society.
#Education and Public Awareness: Educating the general public about the detrimental consequences of corruption fosters a culture of integrity. Raising awareness through campaigns, workshops, and educational programs can discourage individuals from engaging in corrupt activities.&lt;ref&gt;{{Cite web
|url=https://www.oecd.org/governance/ethics/integrity-education.htm
|title=Integrity Education - OECD
|website=Organisation for Economic Co-operation and Development (OECD)
|access-date=2023-08-12
}}&lt;/ref&gt;
#Background Checks and Transparency: Implementing thorough background checks before hiring individuals in sensitive positions can help identify potential risks. Additionally, promoting transparency in recruitment and promotion processes reduces the likelihood of favoritism and bribery.&lt;ref&gt;{{Cite web
|url=https://resources.workable.com/tutorial/employment-background-checks
|title=Employment Background Checks: A Complete Guide
|website=Workable
|access-date=2023-08-12
}}&lt;/ref&gt;
#Regulation and Code of Conduct: Establishing well-structured codes of conduct for organizations sets clear expectations for ethical behavior. By ensuring that rules and regulations are strictly followed, organizations create an environment where corrupt practices are less likely to thrive.&lt;ref&gt;{{Cite web
|url=https://www.sec.gov/Archives/edgar/data/1094007/000119312504044901/dex14.htm
|title=CODE OF BUSINESS CONDUCT AND ETHICS
|website=U.S. Securities and Exchange Commission (SEC)
|access-date=2023-08-12
}}&lt;/ref&gt;
#Accountability and Check-and-Balance Mechanisms: Implementing strong checks and balances within institutions fosters accountability. This includes independent auditing, oversight bodies, and transparent reporting mechanisms. Well-defined rewards for ethical behavior and strict consequences for corruption further reinforce this accountability.&lt;ref&gt;{{Cite web
|url=https://www.premiumtimesng.com/news/top-news/591702-legislation-capacity-gaps-undermining-nigerias-anti-corruption-measures-report.html
|title=Legislation, capacity gaps undermining Nigeria's anti-corruption measures – Report
|website=Premium Times Nigeria
|access-date=2023-08-12
}}&lt;/ref&gt;
#Whistleblower Protection: Encouraging and protecting whistleblowers is crucial. Offering secure channels for reporting corruption allows individuals to come forward without fear of retaliation. Robust legal protections and confidentiality measures safeguard the lives and well-being of those reporting wrongdoing.
#Transparent Procurement Processes: Implementing transparent and competitive procurement processes minimizes opportunities for corruption in public contracting. Open bidding and clear evaluation criteria help prevent favoritism and kickbacks.
#Strengthening Legal Frameworks: Enacting and enforcing comprehensive anti-corruption laws with severe penalties acts as a deterrent. Laws should cover both public and private sectors and ensure that investigations, prosecutions, and trials are carried out effectively.
#Enhancing Technology: Leveraging technology, such as e-governance and digital payment systems, reduces the need for face-to-face interactions and minimizes opportunities for bribery.
#Civil Society and Media Engagement: Engaging civil society organizations and media in monitoring and exposing corrupt practices increases public pressure and ensures that wrongdoers are held accountable.
#Ethics Training and Leadership Development: Providing ethics training for employees and leadership development programs for government officials promote a culture of integrity from the ground up.&lt;ref&gt;{{Cite web|url=https://www.transparency.org/en/what-is-corruption|title=What is corruption?|website=Transparency.org|language=en|access-date=2023-08-12}}&lt;/ref&gt;
===Practice Questions===
#What is the definition of corruption and how does it involve abuse of power?
#What are the two main categories of corruption, and how do they differ?
#How does petty corruption differ from grand corruption in terms of impact?
#Name some major causes of corruption and how they contribute to its occurrence.
#What are the economic, social, and governance effects of corruption?
#How can corruption be prevented through education and public awareness?
#What role does transparency and background checks play in preventing corruption?
#Explain the importance of having well-structured codes of conduct in organizations.
#How can accountability and check-and-balance mechanisms help counter corruption?
#Describe the significance of whistleblower protection in the fight against corruption.

==References==
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I am sure that you probably know, intuitively, about the Real Numbers. However, this resource will disregard intuition of the Real Numbers, and will give a formal, Axiomatic approach to the Real Numbers.

== The Rationals ==
We start off from the rationals, as later we use them to define the Real Numbers. We can define the rationals as

&lt;math&gt;\mathbb{Q} = \{\frac{p}{q} : p,q \in \mathbb{Z}, q \neq 0\}&lt;/math&gt;. We can add, subtract, multiply and divide (except by 0) in the usual way. We call &lt;math&gt;\mathbb{Q}&lt;/math&gt; a [[wikipedia:Field_(mathematics)|field]].

=== On Order ===
I add to this in which the rational numbers are also an [[wikipedia:Ordered field|ordered field]]. That is, a field with a strict order &lt;math&gt;&lt;&lt;/math&gt; such that &lt;math&gt; a &lt; b \rightarrow a + c &lt; b + c&lt;/math&gt;, and also &lt;math&gt;0 &lt; a&lt;/math&gt; and &lt;math&gt;0 &lt; b&lt;/math&gt; implies &lt;math&gt;0 &lt; ab&lt;/math&gt;.

== Limitations of the rational numbers ==
Why do we need the Real numbers at all? This section goes over the limitations of the rationals, which help explain why we need the real numbers.

=== On the Least-upper-bound property ===

==== Definitions ====
We go over the property of Least-upper-bound first. We start by defining the &quot;prerequisite definitions&quot;: Say if we have a subset &lt;math&gt;S&lt;/math&gt; of an ordered field. It is ''bounded above'' if there exists a number &lt;math&gt;b&lt;/math&gt; in said ordered field ''s.t.'' &lt;math&gt;\forall x \in S, b \geq x&lt;/math&gt;. &lt;math&gt;b&lt;/math&gt; is known as an ''upper bound'' of &lt;math&gt;S&lt;/math&gt;.

A ''supremum'' &lt;math&gt;\beta&lt;/math&gt; of &lt;math&gt;S&lt;/math&gt; satisfies 2 properties: It is an upper bound of &lt;math&gt;S&lt;/math&gt;, and all other upper bounds of S are greater than or equal to &lt;math&gt;\beta&lt;/math&gt;. In other words, a supremum of a set is the ''least upper bound'' of a set. As an example, take the set &lt;math&gt;\{x \in \mathbb{N}: x^{-1}, x \neq 0 \}&lt;/math&gt;, in which we denote by &lt;math&gt;F&lt;/math&gt;. It is bounded above, as there exists a number (say &lt;math&gt;\frac{3}{2}&lt;/math&gt;) for which it is greater for all elements in &lt;math&gt;F&lt;/math&gt;, and &lt;math&gt;\frac{3}{2}&lt;/math&gt; is an upper bound. The supremum of &lt;math&gt;F&lt;/math&gt; is 1.

Also, we look back at &lt;math&gt;S&lt;/math&gt;. It is ''bounded below'' if there exists a number &lt;math&gt;c&lt;/math&gt; in said ordered field ''s.t.'' &lt;math&gt;\forall x \in S, c \leq x&lt;/math&gt;, and &lt;math&gt;c&lt;/math&gt; is a ''lower bound.'' The ''infimum,'' or ''greatest lower bound'' of &lt;math&gt;S&lt;/math&gt; is a lower bound of &lt;math&gt;S&lt;/math&gt; that is greater that all other lower bounds of &lt;math&gt;S&lt;/math&gt;.

The property of least-upper-bound then states that any subset &lt;math&gt;X \subseteq Y&lt;/math&gt; (whatever set &lt;math&gt;Y&lt;/math&gt; may be) that is bounded above also has a supremum. The rational numbers do NOT fulfill this property.

'''Theorem. &lt;math&gt;\mathbb{Q}&lt;/math&gt;''' does NOT have the property of Least-upper-bound.

'''Proof.''' We give the example of the subset &lt;math&gt;\{ x \in \mathbb{Q} : x^2 \leq 2 \}&lt;/math&gt;. This subset has an upper bound, with the example of &lt;math&gt;x &gt; 1.5&lt;/math&gt;. However, it does NOT have a supremum, as &lt;math&gt;\sqrt2 \not\in \mathbb{Q}&lt;/math&gt;, even being bounded above. Therefore, '''&lt;math&gt;\mathbb{Q}&lt;/math&gt;''' does NOT have the property of Least-upper-bound. &lt;math&gt;\Box&lt;/math&gt;

=== Completeness ===
The least-upper-bound property is one version of the '''completeness property.''' Intuitively, completeness (of the real numbers) states that there are no &quot;gaps&quot; or &quot;holes&quot; in the real line. Intuitively also, the &quot;holes&quot; in the rational line are equivalent to irrationals. [https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers] There are other ways of defining this property.

==== Cauchy Completeness ====
This definition of completeness states that every [[wikipedia:Cauchy_sequence|Cauchy Sequence]] of numbers in a set converge to another number in a set. (Mainly the real numbers are used in this example.) The Rationals are '''not''' cauchy complete. 

'''Theorem.''' The rational numbers are not Cauchy complete.

'''Proof.''' Take the Cauchy sequence

&lt;math&gt;1, 1.4, 1.41, 1.414 ...&lt;/math&gt;

This sequence is comprised of rationals, but converges to &lt;math&gt;\sqrt2&lt;/math&gt;, but we know that &lt;math&gt;\sqrt2 \not\in \mathbb{Q}&lt;/math&gt;. Therefore, not every Cauchy Sequence of rational numbers converge to another rational number. &lt;math&gt;\Box&lt;/math&gt;


[[Category:Real numbers]]
[[Category:Mathematical analysis]]
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[[File:Strain.jpg|thumb|'''Fig. 1''' This is the starting point for understanding strain as a tensor.  From this one-dimensional system, we see that strain, &lt;math&gt;\Delta L/L,&lt;/math&gt; is inherently dimensionless (length/length). Therefore, the condition for infinitesimal strain, &lt;math&gt;\varepsilon &lt;&lt;1&lt;/math&gt; does not depend on the choice of units.]] This page has two purposes:
 
*To introduce Cauchy's infinitesimal strain at the freshman college level
*To summarize available resources on Wikipedia:

__TOC__
==Wikipedia articles==

{{center|This collection of summaries focuses on finite and infinitesimal strain, but focuses on underestanding Cauchy's strain tensor.}}

===[[Wikipedia:Infinitesimal strain theory]]===
[[File:2D_geometric_strain.svg|thumb|'''Fig. 2''' Visualizing strain in two dimensions]] [[File:Cauchy strain and gradient tensor.svg|thumb|'''Fig. 3''' Cauchy strain and gradient tensor]]
'''[[w:Infinitesimal strain theory|Infinitesimal strain theory]]''' begins by defining Cauchy's strain tensor using a variety of notations. A notation not mentioned is very easy to use when writing by hand. It uses multiple underlines to identify the tensor's [[w:Glossary_of_tensor_theory#Classical_notation|rank]]:

&lt;math display=&quot;block&quot;&gt; \underline\underline\varepsilon\,=\, \tfrac 1 2\left(\underline \nabla\,\underline u(\underline r)+\underline \nabla\,\underline u^T(\underline r)\right)=\underline\nabla\,\underline u_{\,S}(\,\underline r),&lt;/math&gt;

where the number of underlines establishes the rank of the tensor, &quot;T&quot; denotes transpose, and &quot;S&quot; is used to established that the only the symmetric part of the tensor is included. The section '''[[w:Infinitesimal_strain_theory#Geometric_derivation|#Geometric_derivation]]''' uses '''Figure{{Spaces|1}}2''' to help the reader visualize all this. This article also:
:#States transformation rules for &lt;math&gt;\underline\underline\varepsilon&lt;/math&gt; under 3-dimensional rotations, expressed in terms of the basis vectors.&lt;ref&gt;The article has only two references, neither of which allows the casual reader to see the actual text. The rotation matrix expressed in terms of unit vectors is described at https://www1.udel.edu/biology/rosewc/kaap686/notes/matrices_rotations.pdf &lt;/ref&gt;
:#Identifies three invariants under these rotations (including determinant and trace.)
:#Expresses &lt;math&gt;\underline\underline\varepsilon&lt;/math&gt; in cartesian and cylindrical coordinates. 
:#Warns that infinitesimal strain theory is accurate only for very small strains, and is especially likely to yield incorrect results for thin plates and rods.

'''Figure 3''' is not currently on Wikipedia. It illustrates the symmetric and antisymmetric components of &lt;math&gt;\underline\nabla\,\underline u.&lt;/math&gt; The symmetric part rotates the  object so that the deformation's principle axes (&quot;eigenvectors&quot;) are highlighted. The asymmetric part is a rotation that is &quot;pure&quot; only in the limit of small rotations (i.e., where &lt;math&gt;\cos\theta\rightarrow 1.)&lt;/math&gt;

===[[Wikipedia:Deformation (physics)]]===
'''[[w:Deformation (physics)|Deformation (physics)]]''' explains the distinction between a deformation and motion that is only a rigid body displacement as follows: The motion is not a deformation, but merely a rigid body displacement if all possible curves before the deformation maintain their original length after the motion has taken place. The article needs more references, but is noteworthy for its careful and detailed presentation of formulas, many of which clarify of what approximations are required for the transition from general deformations to Cauchy's more tractable infinitesimal (linearized) theory.
*The section '''[[w:Deformation (physics)#Affine_deformation|#Affine deformation]]''' discusses deformations within the context of [[w:Affine transformation|affine transformations.]] 
*Cauchy's strain tensor is based on the partial derivatives of the displacement tensor. The article's discussion  is based on '''[[w:Displacement (geometry)]]''' and '''[[w:Displacement field (mechanics)]]'''.

===[[Wikipedia:Finite strain theory]]===
*''See also [[w:Piola–Kirchhoff stress tensors]], [[w:Reciprocal lattice]], [[w:Einstein notation]], [[w:Covariance and contravariance of vectors]], and [[w:Tensor#As_multidimensional_arrays]]''

'''[[Wikipedia:Finite strain theory|Finite strain theory]]''' is simultaneously essential, as well as a bit too confusing for beginners.  This page resolves a difficulty that plagues a number of other Wikipedia articles: When a small portion of matter moves significantly as it is deformed, which coordinate system do you use?  

The article utilizes covariant notation to almost effortlessly maintain the distinction.  With this notation, both coordinate systems can be used at the same time, and the reader can quickly ascertain which coordinate system is being used.  In its simplest form, we have something like this for vectors:

&lt;math display=&quot;block&quot;&gt;\underline V = \sum_i V_i \hat e^i = \sum_i V^i\hat e_i &lt;/math&gt;

The reader needs to be informed of the nature of &lt;math&gt;\hat e^i&lt;/math&gt; and  &lt;math&gt;\hat e_i&lt;/math&gt; only once.  With tensors of higher rank, you have more than two choices:

&lt;math display=&quot;block&quot;&gt;\underline\underline T = 
\sum_{i,j} T_{j i} \hat e^i\hat e^j = 
\sum_{i,j} T_i^j \hat e_j\hat e^i =
\sum_{i,j} T^{j i} \hat e_i\hat e_j 
&lt;/math&gt;

This notation looks complicated, but is easy to use once you get familiar with it.  This is because in most cases you can dispense with the underlines and unit vectors by utilizing summation notation:

&lt;math display=&quot;block&quot;&gt;V_j=T^j_{\,i}R^i&lt;/math&gt;

Note how a lower summed subscript is always paired with a superscript.  The reader automatically knows all the equivalent variations: &lt;math display=&quot;inline&quot;&gt;V_j=T_{ji}R^i=T_j^{\,i}R_i\,,&lt;/math&gt; and so forth.

This notation is essential in [[w:_Mathematics_of_general_relativity|General Relativity]], and is extremely easy type (requiring only subscripts and superscripts.) A similar notation is also required in [[w:Crystal_system|crystal structures]] where the three spatial basis vectors are not orthogonal, and where the natural basis vectors for displacements are orthogonal to the natural basis vectors for [[w:_Wavenumber|wavenumber]].

===[[Wikipedia:Strain (mechanics)]]===
'''[[w:Strain (mechanics)|Strain  (mechanics)]]''' cryptically defines strain as as the [[w:spatial derivative|spatial derivative]] of [[w:displacement (physics)|displacement]]:
&lt;math display=&quot;block&quot;&gt; \boldsymbol{\varepsilon} \doteq \cfrac{\partial}{\partial\mathbf{X}}\left(\mathbf{x} - \mathbf{X}\right)
 = \boldsymbol{F}'- \boldsymbol{I},&lt;/math&gt;

where {{mvar|'''I'''}} is the [[w:Identity matrix|identity tensor]], where {{math|1='''x''' = '''''F'''''('''X''')}}, and {{math|1= '''''F'''''}} is defined with a link to '''[[w:Finite_strain_theory#Deformation_gradient_tensor|w:Finite strain theory]]'''. 

Other definitions of strain exist: Engineering (Cauchy) strain is, &lt;math&gt;e (L-l)/L,&lt;/math&gt; where &lt;math&gt;l(L)&lt;/math&gt; is the length before (after) deformation. The stretch ratio is &lt;math&gt;1+e&lt;/math&gt;, and the logarithmic strain is &lt;math&gt;\ln(1+e).&lt;/math&gt; Two other definitions of strain (Green and Euler/Alansi) are described at '''[[w:Finite_strain_theory#Finite_strain_tensors|w:Finite strain theory]]'''.

Three types deformation theories are listed:
# '''[[w:Finite strain theory]]''' deals with deformations in which both rotations and strains are arbitrarily large.
# '''[[w:Infinitesimal strain theory]]''' is valid when both strains and rotations are both small. In this case, the undeformed and deformed configurations of the body can be assumed identical.
# ''Large-displacement'' or ''large-rotation theory'' assumes small strains but large rotations and/or displacements.

'''WARNING:''' I question the distinction between #2 and #3 on this list, because I don't see any reason why the rotations and displacements must be small in the linearized theory.  Looking at '''figures 3''' and '''4''', it seems likely that infinitesimal strain theory applies when the rotations are large. My intuition is is that it is important to distinguish between two types of rotation in infinitesimal strain theory. The large scale rotation and displacement in '''figure 4''' is allowed and does not preclude the use of infinitesimal strain theory.  On the other hand, in '''figure 3''', the splitting of &lt;math&gt;\underline\nabla\,\underline u&lt;/math&gt; into a rotation and a deformation (i.e. symmetric and antisymmetric parts), is a different matter.  It must be small, since the &quot;rotation&quot; is only a linear approximation to a true rotation, as can be seen in the approximations &lt;math&gt;\cos\theta\approx 1&lt;/math&gt; and &lt;math&gt;\sin\theta\approx\theta.&lt;/math&gt; The cosine approximation suggests that there is a small (second order) expansion in this so-called &quot;rotation&quot;.  That expansion releases or absorbs energy, and is ignored by the symmetrical strain tensor. Also, '''figure&amp;nbsp;5''' shows how an [[w:Arch|arch]] can be constructed using infinitesimal rotations; the trick is to not excessively bend each square into a trapezoid, and allow the Lagrangian coordinate transformation to do the heavy lifting when it comes to large-scale rotations and deformations.  I'm not saying that the list of three types of deformation theory is wrong, but it is misleading because types #2 and #3 involve almost the same mathematical structure.

The section '''[[w:Strain_(mechanics)#Strain_tensor|#Strain_tensor]]''' reviews material found at '''[[w:Infinitesimal_strain_theory]].'''  The complexity of tensor strain in two or three dimensions are simplified in sections [[w:Strain_(mechanics)#Normal_strain|'''#Normal strain''']], [[w:Strain_(mechanics)#Shear_strain|'''#Shear strain''']], and [[w:Strain_(mechanics)#Volume_strain|'''#Volume strain''']].

[[File:Eulerian versus Lagrangian perspectives illustrated.svg|thumb|275px|'''Fig. 4''' Eulerian versus Lagrangian perspectives illustrated]]
[[File:Strain tensor field for bending.svg|thumb|150px|'''Fig. 5''' Deformation of a square to a trapezoid requires a tensor field]]

===[[Wikipedia:Displacement field (mechanics)]]===
'''[[w:Displacement field (mechanics)|Displacement field (mechanics)]]''' carefully defines the displacement field, beginning with writing the [[w:displacement vector|displacement vector]]. It also addresses the distinction between [[w:Lagrangian and Eulerian specification of the flow field|Lagrangian and Eulerian]] descriptions of the flow of matter. This article is difficult for beginners to follow. For that reason, what follows is a highly simplified attempt to explain what is behind the advanced mathematical language:

'''Figure&amp;nbsp;4''' is designed to informally introduce introductory students to this Lagrangian and Eulerian perspectives: In contrast with most fluids, which can become chaotic, deformations of solid objects are simple (until something breaks!) Figure&amp;nbsp;4 illustrates how it often difficult to follow a fluid element, which suggests that a single stationary reference frame is best suited for the study of fluid dynamics.  In contrast, it is not only easy to follow the path of atoms in a deformation, it is desirable because it permits us to focus on the interaction of these atoms with their nearest neighbors.  It is these nearest neighbor interactions that define how solid matter behaves when deformations occur.

The bending of a two-dimensional &quot;rod&quot; in figure 4 also raises an interesting issue regarding how one visualizes strain (in both two and three dimensions).  In two dimensions, a symmetric tensor has three independent terms. Two of them are &quot;eigenvalues&quot;, or scalars that define how much stretching has occurred in two orthogonal directions. The third term orients these two directions, and since they are orthogonal, a single angle is sufficient to establish their direction (or orientation of the &quot;eigenvectors&quot;.)

[[File:Ellipse 1.svg|25px]] [[File:Rectangle (plain).svg|30px]] [[File:RhombusWhiteBorder.jpg|25px]]  The ellipse, rectangle, and rhombus all serve to represent the eigenvalues of a symmetrical two dimensional matrix.  And all three have analogs in three dimensions. '''Figure&amp;nbsp;5''' establishes that deformation into a trapezoid is not a tensor, but a [[w:Tensor field|tensor field]]: When bending a thin rod or sheet, elements near the outer radius are expanded, while those near the inner radius are compressed.

===[[Wikipedia:Deformation_(engineering)]]===
'''[[Wikipedia:Deformation_(engineering)|Deformation (engineering)]]''' is not a bad article, but the focus is on things like the elastic versus [[w:Plasticity (physics)|plastic regimes]], [[w:Fracture mechanics|fracture]], [[w:Hardening (metallurgy)|strain hardening]], and [[w:Necking (engineering)|necking]]. These are important topics that have little to do with the linear algebra associated with infinitesimal stress and strain.
==Outside the WMF==
*[https://www.brown.edu/Departments/Engineering/Courses/En221/Notes/Kinematics/Kinematics.htm# browwn.edu/Departments/Engineering/Courses/En221] is interesting but not sufficiently clear.
*See also [[Talk:Strain_for_scientists_and_engineers#I'm_still_confused]]

==Footnotes==
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== Probationary_Custodianship ==

We seem to have a bit of conflict between the statement at [[Wikiversity:Candidates_for_Custodianship#Probationary_Custodianship]] and the description of probationary custodianship at [[Wikiversity:Custodianship]]. I don't have strong opinions on this though I do think we should reconcile the two and be clear on the process. I can think of a couple of ways to handle this. One would be to replace probationary custodian with curator as a stepping stone to full custodian. I would also be fine with having the two tracks separate and unconnected. Thoughts? --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 23:44, 18 October 2019 (UTC)

:{{At|Mu301}} We can't have probationary custodianship anymore. According to WMF, a user must be trusted by the community to have access to hidden and deleted content. Probationary custodianship didn't require a vote, and therefore is no longer supported.
:Yes, using Curator as a stepping stone is a good alternative, and as originally proposed was more like probationary custodianship in that a vote wasn't required.
:While I'm at it, I'll mention that I believe any steward who is willing to do vandalism cleanup work at Wikiversity should automatically be granted curator status, so they can feel comfortable in processing the deletes with community support. And I think the curator stepping stone can be bypassed if someone wants to be a custodian and is already a custodian-equivalent or higher on another project. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 01:05, 19 October 2019 (UTC)

::I agree on all three of those points. Let's take a look at the language in the policy page and suggest some changes for the community to discuss. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 16:02, 19 October 2019 (UTC)
::Also, what are your thoughts on mentorship? Is this a practice that we still consider useful? --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 16:16, 19 October 2019 (UTC)
::Re [[User_talk:Vermont#semi-protection|this discussion]]. I think we should make an explicit statement that Global Sysops are welcomed and enoucraged to participate in anti-vandalsim and cleanup. Personally, I didn't realize that there was such hesitancy to act in ways that are obviously beneficial to our community. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 00:14, 3 November 2019 (UTC)
::Please have a look at [[meta:Global sysops]]. By default they will avoid en-wv given that we have both more than 10 custodians and more than 3 are active. We can however hold a discussion to opt-in. I think it is a good idea to do this even if their participation is minimal or occaisonal. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 00:53, 3 November 2019 (UTC)
::Never mind. It looks like we did [https://en.wikiversity.org/w/index.php?title=Wikiversity:Colloquium&amp;curid=28&amp;diff=1055540&amp;oldid=1055440#Opt-in_to_global_sysops opt-in] many years ago. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 01:03, 3 November 2019 (UTC)
== Proposal to allow custodians to use mass-delete ==
{{Archive top}}
{{tracked|T360977}}
At [[special:permalink/2609683#Does_anybody_know_how_to_delete_all_pages_by_a_single_user?]], there was a related discussion about this matter. I brought the agenda to the colloquium for the community's attention, and I was suggested to start a proposal here ([[special:diff/2610994]]). As can be seen at [[Special:ListGroupRights]], only bureaucrats are allowed to use mass-delete under current settings, but many Wikimedia projects allow this to admins (equal to our custodians). Please note that global sysops can also use mass-delete. What does our community think about this? Should we keep the current settings, or should we grant mass-delete to our custodians as a new standard? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 07:12, 8 March 2024 (UTC)
: Curious - is there is a list somewhere of what permissions are provided on which WMF projects? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:43, 8 March 2024 (UTC)
:: [[Special:ListGroupRights]] is available at all WMF projects. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:17, 9 March 2024 (UTC)
:: Please also note that [[:mw:Extension:Nuke]] and [[:b:MediaWiki_Administrator%27s_Handbook/Page_Deletion#Mass_page_deletion]] assumes that admins can handle mass-delete. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:24, 9 March 2024 (UTC)
:a) I'm curious if there's a conflict of interest seeing that [[Wikiversity:Candidates for Custodianship/MathXplore|there is a proposal to make you custodian]] (whatever that means), and b) What's mass-delete? [[User:Username142857|Username142857]] ([[User talk:Username142857|discuss]] • [[Special:Contributions/Username142857|contribs]]) 14:31, 9 March 2024 (UTC)
::It's not a conflict of interest, but the two issues are certainly linked. MathXplore was nominated because they have considerable administrative experience with other Wikis and because they are currently actively involved with Wikversity &quot;cleanup&quot;.  MathXplore's interest in mass-deleting was in response to my request for information on that action.  I am not aware that MathXplore ever asked to be nominated, but if that was the case, I fully support it.  A number of editors are interested in removing low-quality pages (or at least moving them to less conspicuous locations.)  I am somewhat of a bottleneck to that process because as a custodian, I am one of the few people who can delete pages (see '''[[Wikiversity:Deletions]]'''.)  Although there is no conflict of interest, there is a connection between these two requests: MathXplore is less prone to mistakes than I.  Giving MathXplore custodianship and giving custodians mass-deletion would speed up this &quot;cleanup&quot; process.  I have never see MathXplore display poor judgement, and that is why I support giving him both.  Only the adoption of both proposals would allow MathXplore to delete a large number of pages without anybody's approval.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:39, 9 March 2024 (UTC)
:::Thanks @[[User:Username142857|Username142857]] and @[[User:Guy vandegrift|Guy vandegrift]] for pointing out that there is a connection between: (a) this proposal by MathXplore to allow custodians to use mass-delete and (b) the [[Wikiversity:Candidates for Custodianship/MathXplore|nomination (by myself) for MathXplore to move from curator to custodian]]. As Guy points out, it is not necessarily a conflict of interest, but it is important that the connection be pointed out in case anyone has concerns they would like to raise. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:03, 9 March 2024 (UTC)
:::: Thank you for your responses. I noticed that we don't have a COI-related policy. Until formal agreements, [[:w:Wikipedia:Conflict of interest]] can be used as our reference. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:03, 10 March 2024 (UTC)
::::: {{ping|Guy vandegrift}} I declare that I never contacted anyone else about custodianship before [[special:permalink/2611262#Custodianship]]. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:05, 10 March 2024 (UTC)
::::::{{ping|MathXplore}} I'm not accustomed to doing things by the book.  Am I supposed to refrain from voting on this because I do have a clear benefit if it passes? My plan was to be one of the last to vote.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 03:23, 10 March 2024 (UTC)
::::::: Voting or not is all up to you. I'm not sure if you have benefit because the deletions that you are handling are not suitable for mass-delete. They come from different users, and they are neither vandalism nor illegal. Mass-delete aims to accelerate deletion of vandalism etc. by a single user. In the past, I have seen many discussions about admin configurations or deletion policy at other WMF projects (recently I have seen [[:q:special:permalink/3478221#Flood_flag]]), but admins were allowed to join the discussion as one user. So I'm not suprised to see you at here. We are a smaller community (and a GS wiki), if custodians don't join the discussion, then others won't. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 06:16, 10 March 2024 (UTC)
::::::::What does 'illegal' mean in this context, and why couldn't someone just make a resource titled 'Wikiversity: Conflict of interest' to create a COI-related policy? Also, how is this project different from others and what makes a difference meaningful as opposed to not meaningful (examples of both would be nice)? Sorry if I'm asking too many questions :) [[User:Username142857|Username142857]] ([[User talk:Username142857|discuss]] • [[Special:Contributions/Username142857|contribs]]) 15:01, 12 March 2024 (UTC)
::::::::: Examples of illegal content include (but maybe not limited to) copyright violations, privacy violations, serious derogatory, etc. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:04, 13 March 2024 (UTC)
::::::::: Again, we are a smaller community. We don't have enough workforce to make every kind of rules, and making rules is not are main scope (please see [[Wikiversity:Scope]]). You are welcome to make drafts of COI-related policy, but promotion to official policy will need another discussion. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:08, 13 March 2024 (UTC)
::::::::: Unlike others, we are an educational project. Things without educational objectives can disappear ([[WV:CSD]]). Wikipedia is an encyclopedia and may host school projects, but not exactly educational. There are many non-educational content over there, such as video games, movies, animation articles without any educational viewpoints. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:14, 13 March 2024 (UTC)
::::::::: For examples of meaningful differences at Wikiversity, I know [[WikiJournal]] (preprints) and [[Wikidebate]]. WikiJournals can be educational (especially for higher grades), but since they will include original research, they cannot be accepted at Wikipedia ([[:w:Wikipedia:No original research]], [[:w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_manual,_guidebook,_textbook,_or_scientific_journal]]). Wikidebates can help people learn the subject in general or how to debate, but since Wikipedia disallows forums for subjects in general, it cannot exist over there ([[:w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_publisher_of_original_thought]]). [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:22, 13 March 2024 (UTC)
::::::::::How is [[wikipedia:Wikipedia:Wikipediholic|this]] allowed, then? [[User:Username142857|Username142857]] ([[User talk:Username142857|discuss]] • [[Special:Contributions/Username142857|contribs]]) 11:33, 21 March 2024 (UTC)
::::::::::: Things like that can only be agreed upon via local consensus, and that is beyond the judgment of Wikiversity curatorship/custodianship. While &quot;Wikipedia is not a publisher of original thought&quot; (usage of original thought publication in the mainspace will not be accepted in the encyclopedia), some essays might be helpful for others, especially when they are related to the project. Please note that it doesn't mean that essay-only users are always accepted over there ([[:w:WP:NOTHERE]]). [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:26, 21 March 2024 (UTC)
::::::::: An example of possibly non-meaningful differences at Wikiversity was the collection namespace ([[special:permalink/2612077#RFC:_Deprecate_and_remove_the_Collection:_namespace.]]). In the beginning, it was believed to be meaningful, but it was found that it was not used efficiently. Another example of possibly non-meaningful differences at here is the issue that we are facing at here, limiting mass-delete to bureaucrats when more than half of our bureaucrats are inactive. This restriction is affecting the community's ability to handle vandalism quickly, and has nothing to do with Wikiversity's educational objective. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:33, 13 March 2024 (UTC)
:: The first part of your question has been answered by our current custodians, I don't think I have to add anything else. Technical details of mass-delete are provided at [[:mw:Extension:Nuke]]. It can be used by selecting a specific user, and then deleting pages created by them at once. The scope of this feature is to remove obvious vandalism or illegal content (copyright violations, attack pages, etc.) faster. A recent example of using mass-delete is [[Special:Contributions/39.50.199.52]] (already reported to [[WV:RCA]]). As I pointed out above, the Wikimedia standard is to allow mass-delete to admins (they are known as [[Wikiversity:Custodianship|custodians]] in our project). If we are going to make a difference, then those who made them should be able to explain them. I have recently asked why we limited mass-delete to bureaucrats ([[special:diff/2609059]]) and the answer was [[special:diff/2609060]]. We are different from other projects. Meaningful differences should be saved, but questionable differences should be corrected (in this case, restore standard technical settings). We don't need to make differences to be different. This is the background of this proposal. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:01, 10 March 2024 (UTC)

; Voting
: {{support}} It seems that most of the clean-up work at en.wv is currently being performed by sysops/custodians and if they are requesting access to [[Special:Nuke]]  functionality to help them do that work, I'm keen to support that. Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:43, 8 March 2024 (UTC)
: {{support}} There seems to be a lot of content generated and a lot of content left incomplete and dormant.  Probably there's a lot to maintain and I'm all for making it easier on whomever does it. [[User:Addemf|Addemf]] ([[User talk:Addemf|discuss]] • [[Special:Contributions/Addemf|contribs]]) 02:04, 9 March 2024 (UTC)
: {{support}} I've run into this problem where I've wanted to nuke a vandal's contributions but wasn't able to - manually deleting several pages is a hassle. This would be a great additional to our toolkit. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 17:00, 11 March 2024 (UTC)
:{{support}} Safe enough to try, as I believe I wrote on a previous thread. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:40, 11 March 2024 (UTC)
:{{support}} - Would improve response times and any damage from incorrect use is reversible by a custodian anyway. [[User:Elominius|Elominius]] ([[User talk:Elominius|discuss]] • [[Special:Contributions/Elominius|contribs]]) 00:47, 17 March 2024 (UTC)
:{{support}} as it would be very useful in deleting so many pages made by socks of banned users or LTAs. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 20:37, 22 March 2024 (UTC)

{{Archive bottom}}

=== Result ===
I find that there is consensus to allow custodians to make use of the mass delete feature. If someone wants to file the request on phabricator please do so --[[User:DannyS712|DannyS712]] ([[User talk:DannyS712|discuss]] • [[Special:Contributions/DannyS712|contribs]]) 15:17, 25 March 2024 (UTC)
: {{done}}, please see [[:phab:T360977]]. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:33, 26 March 2024 (UTC)
:: (Update) The request has been closed and resolved. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 14:38, 20 May 2024 (UTC)</text>
      <sha1>mo9vwce21y317apatwqevdpf6ogs0h0</sha1>
    </revision>
  </page>
  <page>
    <title>User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells</title>
    <ns>2</ns>
    <id>304065</id>
    <revision>
      <id>2810496</id>
      <parentid>2809480</parentid>
      <timestamp>2026-05-19T19:23:00Z</timestamp>
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        <username>Dc.samizdat</username>
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      <text bytes="144001" sha1="gf47acxpy8rsg0yevq8az5uphol4mu5" xml:space="preserve">= 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 - May 2026}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]] &lt;sub&gt;5&lt;/sub&gt;{3,5,3}&lt;sub&gt;5&lt;/sub&gt;, a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]] {3,5}&lt;sub&gt;5&lt;/sub&gt;, 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.&lt;/blockquote&gt;

== 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=; &quot;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]].&quot;}} 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 &lt;math&gt;2^5&lt;/math&gt; 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 &lt;math&gt;A_5&lt;/math&gt;, 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 &lt;math&gt;A_5&lt;/math&gt; 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 &amp; 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 &amp; 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=; &quot;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 &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

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=; &quot;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, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are Rhombicosidodecahedrons.&quot;}} 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 &quot;Combined hulls&quot; 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''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''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=&quot;''[[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.&quot;}} 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 &quot;Hull # = 8 with 60 vertices&quot;, 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 &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. 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 &lt;math&gt;1&lt;/math&gt; 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 &lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt; &lt;/small&gt;(104.5°), not the 16-cell faces of edge length &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt; (90°), not the 24-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{2} \approx 1.414&lt;/math&gt;&lt;/small&gt; (60°), and not the 600-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}/\phi \approx 0.874&lt;/math&gt;&lt;/small&gt; (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 &lt;math&gt;1&lt;/math&gt;. Truncating at a triangle face of Moxness's Hull #8 exposes a deeper 5-cell triangle face.{{Efn|Each face triangle of edge length &lt;math&gt;1&lt;/math&gt; 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 &lt;math&gt;1&lt;/math&gt; 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 ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; 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 8&lt;sub&gt;3&lt;/sub&gt;, 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=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}} Coxeter was the first to describe the central section 8&lt;sub&gt;3&lt;/sub&gt;, 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 8&lt;sub&gt;3&lt;/sub&gt; 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 4&lt;sub&gt;0&lt;/sub&gt; of the 120-point 600-cell. It is less often noticed that Moxness's Hull #8 rhombicosidodecahedron is the central section 8&lt;sub&gt;3&lt;/sub&gt; 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=; &quot;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.)&quot;}} 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 1&lt;sub&gt;3&lt;/sub&gt;), 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 8&lt;sub&gt;3&lt;/sub&gt;), 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 8&lt;sub&gt;3&lt;/sub&gt; 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 8&lt;sub&gt;3&lt;/sub&gt; 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''&lt;sub&gt;3&lt;/sub&gt; 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 8&lt;sub&gt;3&lt;/sub&gt; 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 1&lt;sub&gt;3&lt;/sub&gt; through 7&lt;sub&gt;3&lt;/sub&gt;, and 60 instances of the central section 8&lt;sub&gt;3&lt;/sub&gt;.{{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.&lt;ref&gt;{{Cite journal|last=Álvarez Elipe|first=Dolores|title=Ensegrities and Tensioned Structures|journal=Journal of Architectural Environment &amp; Structural Engineering Research|date=July 2020|volume=3|issue=3|url=https://www.researchgate.net/publication/343652287_Ensegrities_and_Tensioned_Structures}}&lt;/ref&gt;]]
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 &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Sfn|Coxeter, du Val, Flather &amp; Petrie|1938|p=4|ps=; &quot;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 &quot;[[W:Golden section|golden section]]&quot;. 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]]''.&quot;}} 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 8&lt;sub&gt;3&lt;/sub&gt;) 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=&quot;wikitable floatright&quot; 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]]&lt;BR&gt;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 S&lt;sup&gt;3&lt;/sup&gt;}} 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:&lt;BR&gt;[[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]]&lt;BR&gt;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.&lt;BR&gt;[[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 &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439|ps=; in symmetry group 𝛢&lt;sub&gt;4&lt;/sub&gt; the operation [15]𝑹&lt;sub&gt;q3,q3&lt;/sub&gt; is the 15 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 5-cell; in symmetry group 𝛨&lt;sub&gt;4&lt;/sub&gt; the operation [1200]𝑹&lt;sub&gt;q3,q13&lt;/sub&gt; 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 &lt;math&gt;\phi^2&lt;/math&gt;, 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 &lt;math&gt;\phi^2&lt;/math&gt; {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 &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;-radius 120-cell, the 15.5° {30/1} 120-cell edge chord has length &lt;small&gt;&lt;math&gt;\phi^{-2}&lt;/math&gt;&lt;/small&gt;. The 25.2° {30/2} pentagon face diagonal chord of length &lt;small&gt;&lt;math&gt;\phi^{-1}&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 41.1° 5-cell isocline chord of length &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^2&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 69.8° chord of length &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^3&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 135.5° {30/11} 11-cell edge chord of length &lt;small&gt;&lt;math&gt;\phi^2&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^4&lt;/math&gt;&lt;/small&gt; 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 &lt;small&gt;&lt;math&gt;\sqrt{3}/\phi&lt;/math&gt;&lt;/small&gt;, 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 &lt;math&gt;5^2&lt;/math&gt; 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]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{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 &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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 &quot;reflecting circles&quot;) 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 &lt;math&gt;A_4&lt;/math&gt; 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 &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;B_4&lt;/math&gt; 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 &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

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 &lt;math&gt;F_4&lt;/math&gt; 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 &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

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 &lt;math&gt;H_4&lt;/math&gt; 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 &lt;small&gt;&lt;math&gt;5 \times 14400 / 5 = 14400&lt;/math&gt;&lt;/small&gt;.

=== 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 (&lt;math&gt;A_4&lt;/math&gt; or &lt;math&gt;B_4&lt;/math&gt;).

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 &lt;math&gt;l&lt;/math&gt;, and a distinct rational number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy distinctive flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the rational number &lt;math&gt;h = k/d&lt;/math&gt; and to the polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;d_k&lt;/math&gt; of an &lt;math&gt;n&lt;/math&gt;-gon (with edge &lt;math&gt;d_0=1&lt;/math&gt;) as linear combinations of diagonals.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt;, and to the winding number or density of the polygram (its denominator &lt;math&gt;d&lt;/math&gt;).{{Sfn|Kappraff &amp; Adamson|2004}} The largest &lt;math&gt;k&lt;/math&gt; of any major chord in the 120-cell is 30, and the polygrams &lt;small&gt;&lt;math&gt;\{30/d\}&lt;/math&gt;&lt;/small&gt; 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 &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;. The center section plus either of the smaller sections is &lt;small&gt;&lt;math&gt;\phi = \tfrac{\sqrt{5} + 1}{2} \approx 1.618&lt;/math&gt;&lt;/small&gt;, the larger golden section. Each of the two smaller sections is &lt;small&gt;&lt;math&gt;\Phi = \phi - 1 = \tfrac{1}{\phi} \approx 0.618&lt;/math&gt;&lt;/small&gt;, the smaller golden section.{{Efn|The bitruncated {30/8} chord of the 120-cell provides a geometric derivation of the golden ratio formulas. Consider a 120-cell of radius &lt;small&gt;&lt;math&gt;2\sqrt{2}&lt;/math&gt;&lt;/small&gt; in which the {30/8} chord is &lt;small&gt;&lt;math&gt;2\sqrt{5}&lt;/math&gt;&lt;/small&gt; and the center section of the chord is &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;. Divide results by &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt; to get a radius &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; result. The left section of the chord is:
:&lt;small&gt;&lt;math&gt;\tfrac{\sqrt{5} - 1}{2} \approx 0.618&lt;/math&gt;&lt;/small&gt;
The center section plus the right section is:
:&lt;small&gt;&lt;math&gt;\tfrac{1 + \sqrt{5}}{2} \approx 1.618&lt;/math&gt;&lt;/small&gt;
The sum of these two golden sections is &lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt;&lt;/small&gt;, the chord length.}}

The smaller golden sections &lt;small&gt;&lt;math&gt;\Phi \approx 0.618&lt;/math&gt;&lt;/small&gt; 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 (&quot;Admiral of the fleet&quot;) 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 S&lt;sup&gt;3&lt;/sup&gt;}} 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 8&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #8) rhombicosidodecahedron]]. There are 60 distinct section 8&lt;sub&gt;3&lt;/sub&gt; 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 8&lt;sub&gt;3&lt;/sub&gt; 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 8&lt;sub&gt;3&lt;/sub&gt; 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 6&lt;sub&gt;3&lt;/sub&gt; (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 8&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #8) rhombicosidodecahedron. The 120 semi-regular section 6&lt;sub&gt;3&lt;/sub&gt; 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 S&lt;sub&gt;4&lt;/sub&gt;|ps=; &lt;math&gt;e_2e_3C_{120}\ RID\ P_5\ P_6\ tT&lt;/math&gt;}} These truncated tetrahedra are 1/𝜙 smaller than the 600 cells of the bitruncated 5-cells.

The 60 distinct section 8&lt;sub&gt;3&lt;/sub&gt; 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&lt;ref&gt;{{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}}&lt;/ref&gt; 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.&lt;ref&gt;{{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.}}&lt;/ref&gt;|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 &quot;vector edge cube&quot; (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 8&lt;sub&gt;3&lt;/sub&gt; 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 &lt;math&gt;H_4&lt;/math&gt; 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 &lt;math&gt;F_4&lt;/math&gt; 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 &lt;math&gt;50/25 = 2/1&lt;/math&gt;, the ratio of distinct 12-point Jessen's to distinct 24-point 24-cells is &lt;math&gt;600/225 = 8/3 &lt;/math&gt;.

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 &lt;math&gt;5^2 = 25&lt;/math&gt; distinct 24-cells inscribed in 120 vertices, and is a regular compound of &lt;math&gt;5&lt;/math&gt; 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 &lt;math&gt;5^2 \times 3^2 = 225&lt;/math&gt; distinct 24-cells inscribed in 600 vertices, and is a regular compound of &lt;math&gt;5^2 = 25&lt;/math&gt; 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 ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;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.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

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 &amp; 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 &amp; M | pp=159-166 | issue=May 2007 | url=https://people.eecs.berkeley.edu/~sequin/PAPERS/2007_ISAMA_11Cell.pdf | ref={{SfnRef|Séquin &amp; 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 &amp; 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 &amp; 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 &amp; 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 &amp; 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}}</text>
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  <page>
    <title>24-cell</title>
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        <username>Dc.samizdat</username>
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      <comment>/* Isoclinic rotations */ Corrected hexagram isocline to dodecagram isocline in the text of this section only; footnotes and other sections especially /* Rings */ still require this correction</comment>
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      <text bytes="244071" sha1="pqw267amcsupookv9f026la1hvfey7x" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;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.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
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==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram&lt;sub&gt;2&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges.{{Efn|name=skew hexagram}} All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, forming a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|Because the 24-cell's helical hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram&lt;sub&gt;2&lt;/sub&gt;.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their  isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical hexagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]&lt;sub&gt;2&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}}

Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite &quot;edges&quot; of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram&lt;sub&gt;2&lt;/sub&gt; path. Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew hexagram&lt;sub&gt;2&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram&lt;sub&gt;2&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells.

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt;, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
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|-
!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
|
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; &quot;We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle.&quot;}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}}

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {12}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {1}
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {1}
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[72] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}}


== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="244080" sha1="c9wqhfgjyzyb73zc4utsri671yq3q8k" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;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.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram&lt;sub&gt;2&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges.{{Efn|name=skew hexagram}} All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|Because the 24-cell's helical hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram&lt;sub&gt;2&lt;/sub&gt;.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their  isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical hexagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]&lt;sub&gt;2&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}}

Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite &quot;edges&quot; of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram&lt;sub&gt;2&lt;/sub&gt; path. Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew hexagram&lt;sub&gt;2&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram&lt;sub&gt;2&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells.

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt;, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
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|-
!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
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|
|
|
|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
|
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|
|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; &quot;We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle.&quot;}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}}

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {12}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {1}
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {1}
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[72] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}}


== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

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      <text bytes="2706" sha1="dbxpkb9qpjt6q4o1byzlbyyn50db4fi" xml:space="preserve">{{astronomy}}
'''Andromeda galaxy''' is the nearest major galaxy to the milky way and is cataloged as Messier 31, M31 and NGC 224. Its original name was '''Andromeda Nebula'''&lt;/nowiki&gt; and the name was originated from the wife of Perseus, princess Andromeda from Greek mythology. Andromeda galaxy can be seen with naked eyes, but it is only possible on the moonless nights.
[[File:M31 09-01-2011 (cropped).jpg|thumb|andromeda]]

== Observation history ==
[[File:Andromeda Nebula - Isaac Roberts, 29 December 1888.jpg|120px|thumb|first ever known photo of andromeda nebula]]
The first person ever to discover and describe the Andromeda galaxy with naked eyes were the Persian astronomer Abd al-Rahman al-Sufi around the year 964 CE. He referred it as a &quot;nebulous smear&quot; or &quot;small cloud&quot; in his book. And the first person to give a description of the Andromeda galaxy were German astronomer Simon Marius in year 1612 followed by fellow European astronomer Pierre Louis Maupertuis in 1745. Charles Messier cataloged Andromeda as object M31 in 1764 and incorrectly credited Marius as the discoverer. 

In 1850, the first drawing of Andromeda's spiral structure was made by astronomer William Parsons, 3rd Earl of Rosse.

In the year 1864, William Huggins noted that the spectrum of Andromeda differed from that of a gaseous nebula. Isaac Roberts took one of the first photographs of Andromeda in 1888, which is still commonly thought to be a nebula within our galaxy.

== Formation ==
A major collision of 2 galaxies occurred 2 to 3 billion years ago at the current Andromeda location, which the galaxies' mass ratio were approximately 4. The discovery of a recent merger in the Andromeda galaxy was first based on interpreting its anomalous age-velocity dispersion relation, as well as the fact that 2 billion years ago, star formation throughout Andromeda's disk was much more active than today.

== Structure ==
[[File:Herschel Image of Andromeda Galaxy.jpg|100px|right]]
Based on its appearance in visible light, the Andromeda galaxy can be classified as an SA(s)b galaxy based on the de Vaucouleurs–Sandage extended classification system. However, infrared data from the 2MASS survey and the Spitzer Space Telescope showed that Andromeda is actually a barred spiral galaxy, like the Milky Way. 

== Distance estimate ==
[[File:Andromeda Collides Milky Way.jpg|400px|right|]]
As you might know, space expands constantly. But it's not the case between the Milky way and the Andromeda galaxy. Andromeda galaxy is currently 2.537 million light years away from us. But there's a lot of studies showing that the Milky way and Andromeda is colliding in about 4.5 billion years from now on.


[[Category:Space]]</text>
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    </revision>
  </page>
  <page>
    <title>Talk:One man's look at concept</title>
    <ns>1</ns>
    <id>319076</id>
    <revision>
      <id>2810574</id>
      <parentid>2808402</parentid>
      <timestamp>2026-05-20T09:54:41Z</timestamp>
      <contributor>
        <username>~2026-30014-28</username>
        <id>3078866</id>
      </contributor>
      <comment>/* Жопа */ new section</comment>
      <origin>2810574</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1581" sha1="8e0ekufr033ddr9cxwvda4az50e88p3" xml:space="preserve">== Items to process ==
1) [[W:Fuzzy concept]] should be linked. Potential fuzziness of a concept should perhaps be covered.

2) Operationalization should be covered. I need to clarify whether the result of an operationalization is also called concept (it could be one talks of conceptual level and operational level, but this implies the thing on the operational level is not a concept? Or is it just bad terminology and one should rather talk of vague level and operational level?). I vaguely remember I read about operationalization of the concept of suicide by Durkheim in Sociology by Calhoun et al., for the purpose of sociological research.

3) I could add a section for collocations. That is more of a lexicographical material, but it naturally leads to interesting questions and help tie down the concept of concepts. For instance, are there undefined concepts, are there primitive concepts, etc.

3.1) Add intuitive notion vs. intuitive concept[https://books.google.com/ngrams/graph?content=intuitive+notion%2C+intuitive+concept&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3].

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:42, 28 February 2025 (UTC)
: Expanded. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:16, 28 February 2025 (UTC)

== Жопа ==

Fucking!!! [[Special:Contributions/&amp;#126;2026-30014-28|&amp;#126;2026-30014-28]] ([[User talk:&amp;#126;2026-30014-28|talk]]) 09:54, 20 May 2026 (UTC)</text>
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      <id>2810583</id>
      <parentid>2810574</parentid>
      <timestamp>2026-05-20T10:48:13Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>Reverted edit by [[Special:Contributions/~2026-30014-28|~2026-30014-28]] ([[User_talk:~2026-30014-28|talk]]) to last version by [[User:Atcovi|Atcovi]] using [[Wikiversity:Rollback|rollback]]</comment>
      <origin>2808185</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1421" sha1="b3su7c59opw12w1sovr7s5p4za0e9u3" xml:space="preserve">== Items to process ==
1) [[W:Fuzzy concept]] should be linked. Potential fuzziness of a concept should perhaps be covered.

2) Operationalization should be covered. I need to clarify whether the result of an operationalization is also called concept (it could be one talks of conceptual level and operational level, but this implies the thing on the operational level is not a concept? Or is it just bad terminology and one should rather talk of vague level and operational level?). I vaguely remember I read about operationalization of the concept of suicide by Durkheim in Sociology by Calhoun et al., for the purpose of sociological research.

3) I could add a section for collocations. That is more of a lexicographical material, but it naturally leads to interesting questions and help tie down the concept of concepts. For instance, are there undefined concepts, are there primitive concepts, etc.

3.1) Add intuitive notion vs. intuitive concept[https://books.google.com/ngrams/graph?content=intuitive+notion%2C+intuitive+concept&amp;year_start=1800&amp;year_end=2019&amp;corpus=en-2019&amp;smoothing=3].

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:42, 28 February 2025 (UTC)
: Expanded. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:16, 28 February 2025 (UTC)</text>
      <sha1>b3su7c59opw12w1sovr7s5p4za0e9u3</sha1>
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  </page>
  <page>
    <title>User:Tommy Kronkvist</title>
    <ns>2</ns>
    <id>320737</id>
    <revision>
      <id>2810409</id>
      <parentid>2810153</parentid>
      <timestamp>2026-05-19T12:14:56Z</timestamp>
      <contributor>
        <username>Tommy Kronkvist</username>
        <id>31941</id>
      </contributor>
      <minor />
      <comment>User statistics.</comment>
      <origin>2810409</origin>
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      <text bytes="2645" sha1="dx66058bqfowc83iyjhb1ix5qmqi4w6" xml:space="preserve">&lt;div style=&quot;margin: 0 0 1em 0;&quot;&gt;{{userpage}}&lt;/div&gt;
{{Userboxtop|toptext=Babel:}}
{{#babel:sv|en-4|de-2|la-1}}
{{Userboxbottom}}
[[File:Sorbus torminalis Trunk and canopy.jpg|thumb|310px|The intracanopy of a Wild Service Tree, i.e. &lt;small&gt;''Torminalis glaberrima'' (Gand.) Sennikov &amp; Kurtto, ''Memoranda Soc. Fauna Fl. Fenn.'' 93: 32 (2017).&lt;/small&gt;]]&lt;br /&gt;
 
Most of my wiki contributions are made to [[:species:Main Page|Wikispecies]] where I'm an administrator, bureaucrat and interface admin,&lt;small&gt;&lt;sup&gt;[https://species.wikimedia.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Tommy_Kronkvist (verify)]&lt;/sup&gt;&lt;/small&gt; to the Swedish Wikimedia Chapter [[WMSE:|Wikimedia Sverige]] (WMSE) where I'm an administrator,&lt;small&gt;&lt;sup&gt;(&lt;span class=&quot;plainlinks&quot;&gt;[https://se.wikimedia.org/w/index.php?title=Special:Användare&amp;limit=1&amp;username=Tommy_Kronkvist verify]&lt;/span&gt;)&lt;/sup&gt;&lt;/small&gt; and as administrator and interface administrator at the Swedish version of [[wikivoyage:sv:Huvudsida|Wikivoyage]].&lt;small&gt;&lt;sup&gt;(&lt;span class=&quot;plainlinks&quot;&gt;[https://sv.wikivoyage.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Tommy_Kronkvist verify]&lt;/span&gt;)&lt;/sup&gt;&lt;/small&gt;

So far (May 19, 2026), I've made just over 392,700 edits to 153 of the Wikimedia sister projects&amp;nbsp;– the majority of them to Wikispecies and Wikidata. My global account information for all of Wikimedia can be found [[meta:Special:CentralAuth/Tommy Kronkvist|here]].

Swedish is my mother tongue&amp;nbsp;– even though I was born in Finland&amp;nbsp;– but I feel comfortable speaking and writing English and to some extent in German as well. Odd as it may seem, unfortunately I can't speak any Finnish even though I went to school there for a few years prior to moving to Sweden (see [[w:Swedish-speaking population of Finland|Swedish-speaking population of Finland]] in Wikipedia). I've lived all over Sweden but nowadays reside in Uppsala, the fourth biggest city and former capital of Sweden.

I'm only the fourth generation named &quot;Kronkvist&quot;. My family name consists of two parts: ''kron'' – a short form of the Swedish word ''krona'' meaning 'crown', as in coronation crown or tree crown – and ''kvist'', meaning 'bough' or 'twig'. Hence the name ''Kronkvist'' refers to a twig in the canopy of a forest. I'm the fourth generation of Kronkvist's. Prior to that our family name was ''Mattus'': an oeconym meaning &quot;Matthew's Farm&quot;, dating back to at least 1637.
{{Clear}}
{{User committed identity|a6edd6d2fdbf82621f0cda4e5525c71f8da9b5dfd308242c3c63365e998c32c5406b75448380903265a5403edffd1a0435b61ac943f3c65870db9250f8b884a9|SHA-512|background=#e0e8ff|border=e0e8ff}}</text>
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  <page>
    <title>Einstein Probability Dilation</title>
    <ns>0</ns>
    <id>321584</id>
    <revision>
      <id>2810518</id>
      <parentid>2809410</parentid>
      <timestamp>2026-05-20T00:59:47Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <comment>Adding section on geometric renormalization.</comment>
      <origin>2810518</origin>
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      <text bytes="24157" sha1="jccr58khc8bg7zbb7hd2g7ypsbt5ddj" xml:space="preserve">{{Research project}}
{{To be peer reviewed}}

= Einstein Probability Dilation (EPD) =

== Research abstract ==
'''Einstein Probability Dilation (EPD)''' 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.

EPD 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).

EPD 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 ==
EPD 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]]). EPD 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.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant: &lt;math&gt;Z(P,D)=\int_\Omega D\,dP&lt;/math&gt;.
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure, &lt;math&gt;\mathbb{E}_P[f]=\int_\Omega f\,dP&lt;/math&gt;.

== Probability dilation (reweighting) ==
Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D\,dP}{\int_\Omega D\,dP}\quad\text{for all }A\in\Sigma.&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt; where

&lt;math&gt;\widetilde{p}(x)=\frac{D(x)\,p(x)}{Z},\quad Z=\int_\Omega D(x)\,p(x)\,d\mu.&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

== Example: Lorentz contraction as a positive dilation field ==
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;\gamma(v)=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\quad\text{for } |v| &lt; c.&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;L(v)=\frac{L_0}{\gamma(v)}.&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;, for instance:
* &lt;math&gt;D(x)=\gamma(x)&lt;/math&gt; (weights states by increasing &lt;math&gt;\gamma&lt;/math&gt;), or
* &lt;math&gt;D(x)=\frac{1}{\gamma(x)}&lt;/math&gt; (weights states by decreasing &lt;math&gt;\gamma&lt;/math&gt;),

provided &lt;math&gt;D(x)&lt;/math&gt; is strictly positive over the model’s domain and &lt;math&gt;0&lt;\int_\Omega D\,dP&lt;\infty&lt;/math&gt;.

This example is included purely to illustrate “positive dilation”; it is not presented as a cosmological model.

=== Buffon sampling, the constant \pi, and a Lorentz-induced dilation factor (toy) ===
This brief example shows how &lt;math&gt;\pi&lt;/math&gt; naturally appears from sampling geometry (Buffon’s needle) and how inserting Lorentz contraction into the sampled length produces a strictly positive, dimensionless candidate dilation field. This is included for mathematical intuition; it is not claimed as a physical derivation.

For the classical [[w:Buffon's needle problem|Buffon needle problem]] with needle length &lt;math&gt;L\le d&lt;/math&gt; and line spacing &lt;math&gt;d&lt;/math&gt;, the crossing probability is:
&lt;math&gt;P_{\mathrm{cross}}(L;d)=\frac{2L}{\pi d}.&lt;/math&gt;

If a rest-length &lt;math&gt;L_0&lt;/math&gt; undergoes Lorentz contraction &lt;math&gt;L(v)=L_0/\gamma(v)&lt;/math&gt; for &lt;math&gt;|v| &lt; c&lt;/math&gt;, then the same form gives:
&lt;math&gt;P_{\mathrm{cross}}(v)=\frac{2L_0}{\pi d\,\gamma(v)}.&lt;/math&gt;

Normalizing to the rest case yields a dimensionless sampling ratio:
&lt;math&gt;D_{\pi}(v):=\frac{P_{\mathrm{cross}}(v)}{P_{\mathrm{cross}}(0)}=\frac{1}{\gamma(v)}=\sqrt{1-\frac{v^2}{c^2}}.&lt;/math&gt;

This &lt;math&gt;D_{\pi}(v)&lt;/math&gt; is strictly positive on the domain &lt;math&gt;|v| &lt; c&lt;/math&gt; and can be used (illustratively) as a dilation field in the EPD transform:
&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D_{\pi}(v)\,dP}{\int_\Omega D_{\pi}(v)\,dP}.&lt;/math&gt;

== Theorem-style mathematics expansion ==

=== Definition (EPD reweighting operator) ===
Let &lt;math&gt;P&lt;/math&gt; be a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; and &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; measurable with &lt;math&gt;0&lt;\int_\Omega D\,dP&lt;\infty&lt;/math&gt;. Define &lt;math&gt;\mathrm{EPD}(P;D)=\widetilde{P}&lt;/math&gt; by:

&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D\,dP}{\int_\Omega D\,dP}.&lt;/math&gt;

=== Theorem 1 (Normalization and positivity) ===
&lt;math&gt;\widetilde{P}&lt;/math&gt; is a probability measure. In particular:
# &lt;math&gt;\widetilde{P}(A)\ge 0&lt;/math&gt; for all measurable &lt;math&gt;A&lt;/math&gt;,
# &lt;math&gt;\widetilde{P}(\Omega)=1&lt;/math&gt;,
# If &lt;math&gt;P(A)=0&lt;/math&gt; then &lt;math&gt;\widetilde{P}(A)=0&lt;/math&gt; (absolute continuity preserved).

=== Theorem 2 (Expectation reweighting identity) ===
If &lt;math&gt;f&lt;/math&gt; is integrable under &lt;math&gt;\widetilde{P}&lt;/math&gt;, then

&lt;math&gt;\mathbb{E}_{\widetilde{P}}[f]=\frac{\mathbb{E}_P[D\,f]}{\mathbb{E}_P[D]}.&lt;/math&gt;

=== Theorem 3 (Composition / iteration rule) ===
Let &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D_1)&lt;/math&gt; and &lt;math&gt;\widehat{P}=\mathrm{EPD}(\widetilde{P};D_2)&lt;/math&gt;. Then

&lt;math&gt;\widehat{P}=\mathrm{EPD}(P;D_1D_2)&lt;/math&gt;

provided &lt;math&gt;\int_\Omega D_1D_2\,dP&lt;/math&gt; is finite and nonzero.

=== Theorem 4 (Fixed-point condition) ===
&lt;math&gt;P&lt;/math&gt; is a fixed point of EPD under dilation &lt;math&gt;D&lt;/math&gt; (i.e., &lt;math&gt;\mathrm{EPD}(P;D)=P&lt;/math&gt;) if and only if &lt;math&gt;D&lt;/math&gt; is constant &lt;math&gt;P&lt;/math&gt;-almost everywhere.

== Invariant quantities ==
EPD naturally highlights quantities that are invariant (or transform in controlled ways) under reweighting, including:
* ratios of expectations of the form &lt;math&gt;\mathbb{E}_P[D f]/\mathbb{E}_P[D]&lt;/math&gt;,
* equivalence classes of observables that remain unchanged under specific families of dilations,
* fixed-point structure (trivial dilations) and near-fixed behavior (small perturbations).

= Core axioms of Einstein Probability Dilation =

=== Axiom 1 — Probability primacy ===
Probability measures are treated as primary mathematical objects; structure is inferred from invariant properties of measure transformations.

=== Axiom 2 — Positive dilation ===
Dilation fields are positive (&lt;math&gt;D&lt;/math&gt; is strictly positive) so that reweighting preserves positivity and supports a consistent normalization constant.

=== Axiom 3 — Iterative composability ===
Successive dilations compose multiplicatively (&lt;math&gt;D_1D_2&lt;/math&gt;), enabling multi-step (iterative) dynamics on measures.

= Falsifiability conditions =
A proposed EPD-based model should specify:
* the configuration space &lt;math&gt;\Omega&lt;/math&gt;,
* a baseline measure &lt;math&gt;P&lt;/math&gt;,
* an explicit dilation field &lt;math&gt;D&lt;/math&gt; (with parameters and domain),
* measurable predictions (statistics, scaling laws, invariants) that can be checked against simulation or data,
* a baseline comparison (null model) and evaluation criteria (error measures, goodness-of-fit, or hypothesis tests).

'''Concrete falsifiability test (clustering example):''' Choose a baseline mock galaxy catalog (configuration space &lt;math&gt;\Omega&lt;/math&gt;), define an explicit strictly positive dilation field &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt;, apply EPD-style reweighting (e.g., importance resampling of the empirical measure), and compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt; using a standard estimator (e.g., normalized Landy–Szalay). Define a quantitative success metric (for example, mean-squared error between modeled and observed &lt;math&gt;\xi(r)&lt;/math&gt; over specified separation bins such as &lt;math&gt;r\le 60\,\mathrm{Mpc}/h&lt;/math&gt;). The model is falsified if no parameter choices within the stated domain can meet the pre-registered threshold under the chosen metric, or if the fit fails out-of-sample on withheld bins.

= Numerical simulation and iterative models =

== Simulation model description ==
In discrete demonstrations, the “state space” may be a finite set (e.g., bins, configurations, or catalog points), and the probability measure &lt;math&gt;P&lt;/math&gt; may be represented as a probability vector (or an empirical measure) over those states.

Two equivalent discrete implementations are common:
* '''weighted evaluation''': keep all points and carry weights proportional to &lt;math&gt;D&lt;/math&gt;, or
* '''importance resampling''': draw a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs and comparing \xi(r) ===
This demonstration connects the EPD reweighting idea to a standard observable in large-scale structure: the (monopole) two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;. The goal here is not a precision cosmology fit, but a clean, falsifiable pipeline:

# generate a baseline mock galaxy catalog in a periodic box (approximately unclustered),
# define a strictly positive dilation field &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt; over galaxy positions,
# perform EPD-style importance resampling to obtain a reweighted (clustered) mock catalog,
# measure &lt;math&gt;\xi(r)&lt;/math&gt; using the properly normalized Landy–Szalay estimator,
# tune a small mini-grid over &lt;math&gt;(\sigma,\lambda)&lt;/math&gt; to minimize a simple mismatch score on small scales.

==== Key definitions ====
* Dilation field: &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt;, with &lt;math&gt;\lambda &gt; 0&lt;/math&gt; and &lt;math&gt;\phi(x)\ge 0&lt;/math&gt;.
* Example potential: &lt;math&gt;\phi(x)=\sum_k \exp\!\left(-\|x-s_k\|^2/(2\sigma^2)\right)&lt;/math&gt; using seed points &lt;math&gt;s_k&lt;/math&gt;.
* Landy–Szalay estimator (normalized counts):
  &lt;math&gt;\xi(r)=\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}&lt;/math&gt;,
  where &lt;math&gt;DD,DR,RR&lt;/math&gt; are pair counts normalized by the total number of possible pairs in each catalog.

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
import numpy as np
import matplotlib.pyplot as plt

# ==========================================
# 1) Pair-counting with periodic boundaries
# ==========================================
def pair_counts_weighted(posA, posB, bins, boxsize, wA=None, wB=None,
                         autocorr=False, chunk=512):
    &quot;&quot;&quot;
    Weighted pair counts in radial bins with periodic boundary conditions.
    - autocorr=True: count unique pairs i&lt;j within posA
    - autocorr=False: count all cross pairs between posA and posB
    Returns raw (un-normalized) weighted counts per bin.
    &quot;&quot;&quot;
    posA = np.asarray(posA, float)
    posB = np.asarray(posB, float)
    bins = np.asarray(bins, float)
    nb = len(bins) - 1
    L = float(boxsize)

    if wA is None:
        wA = np.ones(len(posA), float)
    if wB is None:
        wB = np.ones(len(posB), float)
    wA = np.asarray(wA, float)
    wB = np.asarray(wB, float)

    counts = np.zeros(nb, float)

    if autocorr:
        N = len(posA)
        for i0 in range(0, N, chunk):
            i1 = min(N, i0 + chunk)
            A = posA[i0:i1]
            wAi = wA[i0:i1]

            # distances from A to ALL posA (periodic)
            d = A[:, None, :] - posA[None, :, :]
            d -= L * np.round(d / L)
            r = np.sqrt(np.sum(d * d, axis=2))

            W = wAi[:, None] * wA[None, :]

            # accumulate only i&lt;j
            for ii in range(i1 - i0):
                j_start = i0 + ii + 1
                if j_start &gt;= N:
                    continue
                rr = r[ii, j_start:]
                ww = W[ii, j_start:]
                hist, _ = np.histogram(rr, bins=bins, weights=ww)
                counts += hist
        return counts

    else:
        NA = len(posA)
        for i0 in range(0, NA, chunk):
            i1 = min(NA, i0 + chunk)
            A = posA[i0:i1]
            wAi = wA[i0:i1]

            d = A[:, None, :] - posB[None, :, :]
            d -= L * np.round(d / L)
            r = np.sqrt(np.sum(d * d, axis=2))

            W = wAi[:, None] * wB[None, :]
            hist, _ = np.histogram(r, bins=bins, weights=W)
            counts += hist
        return counts


def landy_szalay_xi(DD_raw, DR_raw, RR_raw, nD, nR):
    &quot;&quot;&quot;
    Properly normalized Landy–Szalay estimator:
      DD = DD_raw / [nD (nD-1)/2]
      DR = DR_raw / [nD nR]
      RR = RR_raw / [nR (nR-1)/2]
      xi = (DD - 2 DR + RR) / RR
    &quot;&quot;&quot;
    DD = DD_raw / (nD * (nD - 1) / 2.0)
    DR = DR_raw / (nD * nR)
    RR = RR_raw / (nR * (nR - 1) / 2.0)
    return (DD - 2 * DR + RR) / np.maximum(RR, 1e-30)


# ==========================================
# 2) EPD-style dilation field + resampling
# ==========================================
def gaussian_seed_field(pos, seeds, boxsize, sigma):
    &quot;&quot;&quot;
    phi(x) = sum_k exp(-||x-seed_k||^2/(2 sigma^2)) with periodic distances.
    Always nonnegative.
    &quot;&quot;&quot;
    pos = np.asarray(pos, float)
    seeds = np.asarray(seeds, float)
    L = float(boxsize)

    d = pos[:, None, :] - seeds[None, :, :]
    d -= L * np.round(d / L)
    r2 = np.sum(d * d, axis=2)
    phi = np.exp(-0.5 * r2 / (sigma ** 2)).sum(axis=1)
    return phi


def epd_resample_catalog(pos0, phi, lam, rng):
    &quot;&quot;&quot;
    Importance-resampling version of discrete EPD on an empirical measure:
      weights ∝ D(x)=exp(lam * phi(x))
      resample points with prob ∝ weights
    &quot;&quot;&quot;
    logw = lam * phi
    logw -= np.max(logw)          # stabilize
    w = np.exp(logw)              # strictly positive
    p = w / w.sum()
    idx = rng.choice(len(pos0), size=len(pos0), replace=True, p=p)
    return pos0[idx]


# ==========================================
# 3) Load real xi0(r) for reference (BOSS)
# ==========================================
# NOTE: Set this path to wherever you saved the file locally.
# The example below matches the earlier DR11 multipoles file name.
path = &quot;samushia_2013_CMASSDR11_xi_multipoles.dat&quot;
vals = np.loadtxt(path)
xi_obs = vals[:16]  # monopole xi0

# BOSS bins: edges 24..152 step 8 =&gt; centers 28..148
bins = np.arange(24, 152 + 8, 8)
r_centers = (bins[:-1] + bins[1:]) / 2


# ==========================================
# 4) Build baseline mock + randoms (periodic box)
# ==========================================
rng = np.random.default_rng(7)

boxsize = 200.0
N_data0 = 1400
N_rand = 4500

pos0 = rng.uniform(0, boxsize, size=(N_data0, 3))
posR = rng.uniform(0, boxsize, size=(N_rand, 3))

# Precompute RR once (random-random doesn't change during grid search)
RR = pair_counts_weighted(posR, posR, bins, boxsize, autocorr=True, chunk=512)


# ==========================================
# 5) Define seed locations (kept fixed during tuning)
# ==========================================
K = 12
seeds = rng.uniform(0, boxsize, size=(K, 3))


# ==========================================
# 6) Objective: simple &quot;success&quot; on small-r bins
# ==========================================
def mse_on_bins(xi_model, xi_target, r_centers, rmax=60.0):
    &quot;&quot;&quot;
    Mean squared error over bins with r &lt;= rmax (simple success criterion).
    &quot;&quot;&quot;
    m = r_centers &lt;= rmax
    return float(np.mean((xi_model[m] - xi_target[m]) ** 2))


# ==========================================
# 7) Mini-grid search over (lambda, sigma)
# ==========================================
sigmas = [6.0, 8.0, 10.0, 12.0, 15.0]   # seed width (correlation length scale)
lams   = [0.5, 1.0, 2.0, 3.0, 4.0, 5.0] # dilation strength

best = None

# Baseline xi (should be ~0)
DD0 = pair_counts_weighted(pos0, pos0, bins, boxsize, autocorr=True, chunk=512)
DR0 = pair_counts_weighted(pos0, posR, bins, boxsize, autocorr=False, chunk=512)
xi_base = landy_szalay_xi(DD0, DR0, RR, nD=len(pos0), nR=len(posR))

for sigma in sigmas:
    phi = gaussian_seed_field(pos0, seeds, boxsize, sigma)

    for lam in lams:
        pos1 = epd_resample_catalog(pos0, phi, lam, rng)

        DD1 = pair_counts_weighted(pos1, pos1, bins, boxsize, autocorr=True, chunk=512)
        DR1 = pair_counts_weighted(pos1, posR, bins, boxsize, autocorr=False, chunk=512)
        xi_epd = landy_szalay_xi(DD1, DR1, RR, nD=len(pos1), nR=len(posR))

        score = mse_on_bins(xi_epd, xi_obs, r_centers, rmax=60.0)

        if best is None or score &lt; best[0]:
            best = (score, sigma, lam, xi_epd)

best_score, best_sigma, best_lam, xi_best = best

# ==========================================
# 8) Plot baseline vs best EPD vs observed
# ==========================================
plt.figure()
plt.plot(r_centers, xi_base, marker='o', label=&quot;Mock baseline (uniform Poisson)&quot;)
plt.plot(r_centers, xi_best, marker='o',
         label=f&quot;Best EPD-resampled (sigma={best_sigma}, lambda={best_lam})&quot;)
plt.plot(r_centers, xi_obs, marker='o', linestyle='--',
         label=&quot;Observed BOSS DR11 ξ₀(r) (reference)&quot;)
plt.axhline(0, linewidth=1)
plt.xlabel(&quot;r [Mpc/h]&quot;)
plt.ylabel(&quot;ξ(r) (monopole estimate)&quot;)
plt.title(&quot;EPD reweighting of mock catalogs: mini-grid tuning (toy demonstration)&quot;)
plt.legend()
plt.show()

print(&quot;=== Mini-grid search result ===&quot;)
print(f&quot;Best score (MSE on r&lt;=60): {best_score:.6e}&quot;)
print(f&quot;Best sigma: {best_sigma}  Best lambda: {best_lam}&quot;)
&lt;/syntaxhighlight&gt;

'''How to run:''' download the BOSS multipoles text file to your computer and set &lt;code&gt;path&lt;/code&gt; in the script to that local filename. Then run the script in Python (NumPy + Matplotlib required). The baseline curve should lie near &lt;math&gt;\xi\approx 0&lt;/math&gt;, while the EPD-resampled mock typically shows positive small-scale clustering.

= Scope and limitations =
EPD is a mathematical framework for measure transformations. It does not claim:
* a replacement theory for GR/QM,
* empirical confirmation without explicit predictions and tests.

== 8. Speculative Extensions and Geometric Renormalization ==
''The following section is exploratory and speculative. It proposes possible extensions of Einstein Probability Dilation (EPD) into probabilistic geometry, curvature-weighted path integrals, and geometric renormalization. These ideas are heuristic and are not presently established as formal derivations or experimentally verified physical models.''

Recent mathematical work extending Buffon’s needle problem to curved manifolds has suggested a possible connection between probability distributions and intrinsic geometry. In particular, studies of “Buffon deficits” on Gaussian manifolds demonstrated that deviations from the classical flat-space Buffon probability may encode Gaussian curvature through second-order geometric corrections.

In the classical flat Buffon problem, the expected crossing probability approaches:

[

\frac{2}{\pi}

]

for sufficiently small needles. On curved manifolds, however, the crossing probability acquires curvature-dependent corrections. In the small-length limit, these deviations may be interpreted as probabilistic signatures of intrinsic geometry.

This observation motivates a speculative extension of Einstein Probability Dilation (EPD): spacetime curvature may not only influence probability distributions, but probability structure itself may participate in the emergence or modulation of geometric behavior.

Within this interpretation, EPD may be viewed as a probabilistic-geometric framework in which curvature acts as a statistical weighting mechanism on path behavior. Rather than treating quantum path integrals strictly as sums over equally admissible histories, the geometry of spacetime could modulate the statistical contribution of classes of paths through curvature-dependent probability measures.

A possible schematic form is:

[

\int \mathcal{D}[x];W(K);e^{iS[x]/\hbar},

]

where:

* (e^{iS[x]/\hbar}) represents the standard quantum probability amplitude,
* (W(K)) represents a curvature-dependent geometric weighting factor,
* (K) denotes local Gaussian or spacetime curvature.

Under this interpretation, the two probabilistic structures are not identical in mathematical character.

The Feynman term:

[

e^{iS/\hbar}

]

is a complex probability amplitude capable of quantum interference and phase cancellation, whereas the curvature-weighting term behaves more like a real geometric measure density acting on the statistical structure of path space itself.

Accordingly, EPD may not “mix” probabilities directly, but instead combine:

* a quantum interference structure,
* with a curvature-dependent probabilistic geometry.

The Buffon deficit framework suggests that curvature may appear as a probabilistic deviation from flat geometric behavior. In the EPD interpretation, this motivates the possibility that curvature-weighted probability measures could alter the statistical admissibility of quantum paths at extremely small scales.

If such curvature-weighting naturally suppresses pathological ultraviolet path contributions, the framework could potentially resemble a form of geometric renormalization in which divergence control emerges from geometric probability structure rather than purely algebraic subtraction procedures.

A speculative expansion of the geometric weighting term might take the form:

[

W(K,L) \approx 1 + cKL^2 + O(L^3),

]

where:

* (L) represents a characteristic microscopic path scale,
* (K) represents local curvature,
* (c) is a proportionality constant.

Such a correction would recover ordinary flat-space behavior in the limit:

[

K \rightarrow 0.

]

At present these ideas remain heuristic and conceptual. However, they suggest a possible bridge between:

* stochastic geometry,
* probabilistic curvature,
* quantum path integrals,
* and emergent spacetime structure.

=== Open Questions ===

* Can curvature-weighted path measures be formally derived from probabilistic geometry?
* Can Buffon deficit relations be generalized from Gaussian manifolds to relativistic spacetime manifolds?
* Does curvature-weighting naturally regulate ultraviolet divergences in quantum field theory?
* What is the relationship between Gaussian curvature corrections and spacetime curvature tensors in General Relativity?
* Could probabilistic-geometric weighting contribute to emergent spacetime models or stochastic gravity frameworks?

= 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.

= See also =
* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Cosmological principle|Cosmological principle]]
* [[w:Holographic principle|Holographic principle]]
* [[w:Fractal geometry|Fractal geometry]]
* [[w:Quantum field theory|Quantum field theory]]
* [[w:Langlands program|Langlands program]]

= References =
==== Probability / measure theory ====
* [[w:Measure (mathematics)|Measure (mathematics)]]
* [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]]
* [[w:Probability theory|Probability theory]]

==== Computing / simulation ====
* [[w:Monte Carlo method|Monte Carlo method]]
* [[w:Importance sampling|Importance sampling]]

== Copyright and licensing ==
This Wikiversity page is authored by Howard Richardson. Please attribute appropriately when reusing or adapting this material.

Earlier and related versions may also be available on OSF and arXiv under the licenses stated on those platforms. Where multiple versions exist, the license shown on the specific platform/version governs reuse of that version.</text>
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      <timestamp>2026-05-20T01:14:19Z</timestamp>
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        <id>2995240</id>
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      <comment>EBP and EPD clarification.</comment>
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      <text bytes="24438" sha1="23ap38gc67tqnn0n9s6eapgtzq1hg9j" xml:space="preserve">{{Research project}}
{{To be peer reviewed}}

= Einstein Probability Dilation (EPD) =

== Research abstract ==
'''Einstein Probability Dilation (EPD)''' 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.

EPD 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).

EPD 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 ==
EPD 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]]). EPD 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.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant: &lt;math&gt;Z(P,D)=\int_\Omega D\,dP&lt;/math&gt;.
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure, &lt;math&gt;\mathbb{E}_P[f]=\int_\Omega f\,dP&lt;/math&gt;.

== Probability dilation (reweighting) ==
Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D\,dP}{\int_\Omega D\,dP}\quad\text{for all }A\in\Sigma.&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt; where

&lt;math&gt;\widetilde{p}(x)=\frac{D(x)\,p(x)}{Z},\quad Z=\int_\Omega D(x)\,p(x)\,d\mu.&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

== Example: Lorentz contraction as a positive dilation field ==
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;\gamma(v)=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\quad\text{for } |v| &lt; c.&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;L(v)=\frac{L_0}{\gamma(v)}.&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;, for instance:
* &lt;math&gt;D(x)=\gamma(x)&lt;/math&gt; (weights states by increasing &lt;math&gt;\gamma&lt;/math&gt;), or
* &lt;math&gt;D(x)=\frac{1}{\gamma(x)}&lt;/math&gt; (weights states by decreasing &lt;math&gt;\gamma&lt;/math&gt;),

provided &lt;math&gt;D(x)&lt;/math&gt; is strictly positive over the model’s domain and &lt;math&gt;0&lt;\int_\Omega D\,dP&lt;\infty&lt;/math&gt;.

This example is included purely to illustrate “positive dilation”; it is not presented as a cosmological model.

=== Buffon sampling, the constant \pi, and a Lorentz-induced dilation factor (toy) ===
This brief example shows how &lt;math&gt;\pi&lt;/math&gt; naturally appears from sampling geometry (Buffon’s needle) and how inserting Lorentz contraction into the sampled length produces a strictly positive, dimensionless candidate dilation field. This is included for mathematical intuition; it is not claimed as a physical derivation.

For the classical [[w:Buffon's needle problem|Buffon needle problem]] with needle length &lt;math&gt;L\le d&lt;/math&gt; and line spacing &lt;math&gt;d&lt;/math&gt;, the crossing probability is:
&lt;math&gt;P_{\mathrm{cross}}(L;d)=\frac{2L}{\pi d}.&lt;/math&gt;

If a rest-length &lt;math&gt;L_0&lt;/math&gt; undergoes Lorentz contraction &lt;math&gt;L(v)=L_0/\gamma(v)&lt;/math&gt; for &lt;math&gt;|v| &lt; c&lt;/math&gt;, then the same form gives:
&lt;math&gt;P_{\mathrm{cross}}(v)=\frac{2L_0}{\pi d\,\gamma(v)}.&lt;/math&gt;

Normalizing to the rest case yields a dimensionless sampling ratio:
&lt;math&gt;D_{\pi}(v):=\frac{P_{\mathrm{cross}}(v)}{P_{\mathrm{cross}}(0)}=\frac{1}{\gamma(v)}=\sqrt{1-\frac{v^2}{c^2}}.&lt;/math&gt;

This &lt;math&gt;D_{\pi}(v)&lt;/math&gt; is strictly positive on the domain &lt;math&gt;|v| &lt; c&lt;/math&gt; and can be used (illustratively) as a dilation field in the EPD transform:
&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D_{\pi}(v)\,dP}{\int_\Omega D_{\pi}(v)\,dP}.&lt;/math&gt;

== Theorem-style mathematics expansion ==

=== Definition (EPD reweighting operator) ===
Let &lt;math&gt;P&lt;/math&gt; be a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; and &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; measurable with &lt;math&gt;0&lt;\int_\Omega D\,dP&lt;\infty&lt;/math&gt;. Define &lt;math&gt;\mathrm{EPD}(P;D)=\widetilde{P}&lt;/math&gt; by:

&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D\,dP}{\int_\Omega D\,dP}.&lt;/math&gt;

=== Theorem 1 (Normalization and positivity) ===
&lt;math&gt;\widetilde{P}&lt;/math&gt; is a probability measure. In particular:
# &lt;math&gt;\widetilde{P}(A)\ge 0&lt;/math&gt; for all measurable &lt;math&gt;A&lt;/math&gt;,
# &lt;math&gt;\widetilde{P}(\Omega)=1&lt;/math&gt;,
# If &lt;math&gt;P(A)=0&lt;/math&gt; then &lt;math&gt;\widetilde{P}(A)=0&lt;/math&gt; (absolute continuity preserved).

=== Theorem 2 (Expectation reweighting identity) ===
If &lt;math&gt;f&lt;/math&gt; is integrable under &lt;math&gt;\widetilde{P}&lt;/math&gt;, then

&lt;math&gt;\mathbb{E}_{\widetilde{P}}[f]=\frac{\mathbb{E}_P[D\,f]}{\mathbb{E}_P[D]}.&lt;/math&gt;

=== Theorem 3 (Composition / iteration rule) ===
Let &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D_1)&lt;/math&gt; and &lt;math&gt;\widehat{P}=\mathrm{EPD}(\widetilde{P};D_2)&lt;/math&gt;. Then

&lt;math&gt;\widehat{P}=\mathrm{EPD}(P;D_1D_2)&lt;/math&gt;

provided &lt;math&gt;\int_\Omega D_1D_2\,dP&lt;/math&gt; is finite and nonzero.

=== Theorem 4 (Fixed-point condition) ===
&lt;math&gt;P&lt;/math&gt; is a fixed point of EPD under dilation &lt;math&gt;D&lt;/math&gt; (i.e., &lt;math&gt;\mathrm{EPD}(P;D)=P&lt;/math&gt;) if and only if &lt;math&gt;D&lt;/math&gt; is constant &lt;math&gt;P&lt;/math&gt;-almost everywhere.

== Invariant quantities ==
EPD naturally highlights quantities that are invariant (or transform in controlled ways) under reweighting, including:
* ratios of expectations of the form &lt;math&gt;\mathbb{E}_P[D f]/\mathbb{E}_P[D]&lt;/math&gt;,
* equivalence classes of observables that remain unchanged under specific families of dilations,
* fixed-point structure (trivial dilations) and near-fixed behavior (small perturbations).

= Core axioms of Einstein Probability Dilation =

=== Axiom 1 — Probability primacy ===
Probability measures are treated as primary mathematical objects; structure is inferred from invariant properties of measure transformations.

=== Axiom 2 — Positive dilation ===
Dilation fields are positive (&lt;math&gt;D&lt;/math&gt; is strictly positive) so that reweighting preserves positivity and supports a consistent normalization constant.

=== Axiom 3 — Iterative composability ===
Successive dilations compose multiplicatively (&lt;math&gt;D_1D_2&lt;/math&gt;), enabling multi-step (iterative) dynamics on measures.

= Falsifiability conditions =
A proposed EPD-based model should specify:
* the configuration space &lt;math&gt;\Omega&lt;/math&gt;,
* a baseline measure &lt;math&gt;P&lt;/math&gt;,
* an explicit dilation field &lt;math&gt;D&lt;/math&gt; (with parameters and domain),
* measurable predictions (statistics, scaling laws, invariants) that can be checked against simulation or data,
* a baseline comparison (null model) and evaluation criteria (error measures, goodness-of-fit, or hypothesis tests).

'''Concrete falsifiability test (clustering example):''' Choose a baseline mock galaxy catalog (configuration space &lt;math&gt;\Omega&lt;/math&gt;), define an explicit strictly positive dilation field &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt;, apply EPD-style reweighting (e.g., importance resampling of the empirical measure), and compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt; using a standard estimator (e.g., normalized Landy–Szalay). Define a quantitative success metric (for example, mean-squared error between modeled and observed &lt;math&gt;\xi(r)&lt;/math&gt; over specified separation bins such as &lt;math&gt;r\le 60\,\mathrm{Mpc}/h&lt;/math&gt;). The model is falsified if no parameter choices within the stated domain can meet the pre-registered threshold under the chosen metric, or if the fit fails out-of-sample on withheld bins.

= Numerical simulation and iterative models =

== Simulation model description ==
In discrete demonstrations, the “state space” may be a finite set (e.g., bins, configurations, or catalog points), and the probability measure &lt;math&gt;P&lt;/math&gt; may be represented as a probability vector (or an empirical measure) over those states.

Two equivalent discrete implementations are common:
* '''weighted evaluation''': keep all points and carry weights proportional to &lt;math&gt;D&lt;/math&gt;, or
* '''importance resampling''': draw a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs and comparing \xi(r) ===
This demonstration connects the EPD reweighting idea to a standard observable in large-scale structure: the (monopole) two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;. The goal here is not a precision cosmology fit, but a clean, falsifiable pipeline:

# generate a baseline mock galaxy catalog in a periodic box (approximately unclustered),
# define a strictly positive dilation field &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt; over galaxy positions,
# perform EPD-style importance resampling to obtain a reweighted (clustered) mock catalog,
# measure &lt;math&gt;\xi(r)&lt;/math&gt; using the properly normalized Landy–Szalay estimator,
# tune a small mini-grid over &lt;math&gt;(\sigma,\lambda)&lt;/math&gt; to minimize a simple mismatch score on small scales.

==== Key definitions ====
* Dilation field: &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt;, with &lt;math&gt;\lambda &gt; 0&lt;/math&gt; and &lt;math&gt;\phi(x)\ge 0&lt;/math&gt;.
* Example potential: &lt;math&gt;\phi(x)=\sum_k \exp\!\left(-\|x-s_k\|^2/(2\sigma^2)\right)&lt;/math&gt; using seed points &lt;math&gt;s_k&lt;/math&gt;.
* Landy–Szalay estimator (normalized counts):
  &lt;math&gt;\xi(r)=\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}&lt;/math&gt;,
  where &lt;math&gt;DD,DR,RR&lt;/math&gt; are pair counts normalized by the total number of possible pairs in each catalog.

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
import numpy as np
import matplotlib.pyplot as plt

# ==========================================
# 1) Pair-counting with periodic boundaries
# ==========================================
def pair_counts_weighted(posA, posB, bins, boxsize, wA=None, wB=None,
                         autocorr=False, chunk=512):
    &quot;&quot;&quot;
    Weighted pair counts in radial bins with periodic boundary conditions.
    - autocorr=True: count unique pairs i&lt;j within posA
    - autocorr=False: count all cross pairs between posA and posB
    Returns raw (un-normalized) weighted counts per bin.
    &quot;&quot;&quot;
    posA = np.asarray(posA, float)
    posB = np.asarray(posB, float)
    bins = np.asarray(bins, float)
    nb = len(bins) - 1
    L = float(boxsize)

    if wA is None:
        wA = np.ones(len(posA), float)
    if wB is None:
        wB = np.ones(len(posB), float)
    wA = np.asarray(wA, float)
    wB = np.asarray(wB, float)

    counts = np.zeros(nb, float)

    if autocorr:
        N = len(posA)
        for i0 in range(0, N, chunk):
            i1 = min(N, i0 + chunk)
            A = posA[i0:i1]
            wAi = wA[i0:i1]

            # distances from A to ALL posA (periodic)
            d = A[:, None, :] - posA[None, :, :]
            d -= L * np.round(d / L)
            r = np.sqrt(np.sum(d * d, axis=2))

            W = wAi[:, None] * wA[None, :]

            # accumulate only i&lt;j
            for ii in range(i1 - i0):
                j_start = i0 + ii + 1
                if j_start &gt;= N:
                    continue
                rr = r[ii, j_start:]
                ww = W[ii, j_start:]
                hist, _ = np.histogram(rr, bins=bins, weights=ww)
                counts += hist
        return counts

    else:
        NA = len(posA)
        for i0 in range(0, NA, chunk):
            i1 = min(NA, i0 + chunk)
            A = posA[i0:i1]
            wAi = wA[i0:i1]

            d = A[:, None, :] - posB[None, :, :]
            d -= L * np.round(d / L)
            r = np.sqrt(np.sum(d * d, axis=2))

            W = wAi[:, None] * wB[None, :]
            hist, _ = np.histogram(r, bins=bins, weights=W)
            counts += hist
        return counts


def landy_szalay_xi(DD_raw, DR_raw, RR_raw, nD, nR):
    &quot;&quot;&quot;
    Properly normalized Landy–Szalay estimator:
      DD = DD_raw / [nD (nD-1)/2]
      DR = DR_raw / [nD nR]
      RR = RR_raw / [nR (nR-1)/2]
      xi = (DD - 2 DR + RR) / RR
    &quot;&quot;&quot;
    DD = DD_raw / (nD * (nD - 1) / 2.0)
    DR = DR_raw / (nD * nR)
    RR = RR_raw / (nR * (nR - 1) / 2.0)
    return (DD - 2 * DR + RR) / np.maximum(RR, 1e-30)


# ==========================================
# 2) EPD-style dilation field + resampling
# ==========================================
def gaussian_seed_field(pos, seeds, boxsize, sigma):
    &quot;&quot;&quot;
    phi(x) = sum_k exp(-||x-seed_k||^2/(2 sigma^2)) with periodic distances.
    Always nonnegative.
    &quot;&quot;&quot;
    pos = np.asarray(pos, float)
    seeds = np.asarray(seeds, float)
    L = float(boxsize)

    d = pos[:, None, :] - seeds[None, :, :]
    d -= L * np.round(d / L)
    r2 = np.sum(d * d, axis=2)
    phi = np.exp(-0.5 * r2 / (sigma ** 2)).sum(axis=1)
    return phi


def epd_resample_catalog(pos0, phi, lam, rng):
    &quot;&quot;&quot;
    Importance-resampling version of discrete EPD on an empirical measure:
      weights ∝ D(x)=exp(lam * phi(x))
      resample points with prob ∝ weights
    &quot;&quot;&quot;
    logw = lam * phi
    logw -= np.max(logw)          # stabilize
    w = np.exp(logw)              # strictly positive
    p = w / w.sum()
    idx = rng.choice(len(pos0), size=len(pos0), replace=True, p=p)
    return pos0[idx]


# ==========================================
# 3) Load real xi0(r) for reference (BOSS)
# ==========================================
# NOTE: Set this path to wherever you saved the file locally.
# The example below matches the earlier DR11 multipoles file name.
path = &quot;samushia_2013_CMASSDR11_xi_multipoles.dat&quot;
vals = np.loadtxt(path)
xi_obs = vals[:16]  # monopole xi0

# BOSS bins: edges 24..152 step 8 =&gt; centers 28..148
bins = np.arange(24, 152 + 8, 8)
r_centers = (bins[:-1] + bins[1:]) / 2


# ==========================================
# 4) Build baseline mock + randoms (periodic box)
# ==========================================
rng = np.random.default_rng(7)

boxsize = 200.0
N_data0 = 1400
N_rand = 4500

pos0 = rng.uniform(0, boxsize, size=(N_data0, 3))
posR = rng.uniform(0, boxsize, size=(N_rand, 3))

# Precompute RR once (random-random doesn't change during grid search)
RR = pair_counts_weighted(posR, posR, bins, boxsize, autocorr=True, chunk=512)


# ==========================================
# 5) Define seed locations (kept fixed during tuning)
# ==========================================
K = 12
seeds = rng.uniform(0, boxsize, size=(K, 3))


# ==========================================
# 6) Objective: simple &quot;success&quot; on small-r bins
# ==========================================
def mse_on_bins(xi_model, xi_target, r_centers, rmax=60.0):
    &quot;&quot;&quot;
    Mean squared error over bins with r &lt;= rmax (simple success criterion).
    &quot;&quot;&quot;
    m = r_centers &lt;= rmax
    return float(np.mean((xi_model[m] - xi_target[m]) ** 2))


# ==========================================
# 7) Mini-grid search over (lambda, sigma)
# ==========================================
sigmas = [6.0, 8.0, 10.0, 12.0, 15.0]   # seed width (correlation length scale)
lams   = [0.5, 1.0, 2.0, 3.0, 4.0, 5.0] # dilation strength

best = None

# Baseline xi (should be ~0)
DD0 = pair_counts_weighted(pos0, pos0, bins, boxsize, autocorr=True, chunk=512)
DR0 = pair_counts_weighted(pos0, posR, bins, boxsize, autocorr=False, chunk=512)
xi_base = landy_szalay_xi(DD0, DR0, RR, nD=len(pos0), nR=len(posR))

for sigma in sigmas:
    phi = gaussian_seed_field(pos0, seeds, boxsize, sigma)

    for lam in lams:
        pos1 = epd_resample_catalog(pos0, phi, lam, rng)

        DD1 = pair_counts_weighted(pos1, pos1, bins, boxsize, autocorr=True, chunk=512)
        DR1 = pair_counts_weighted(pos1, posR, bins, boxsize, autocorr=False, chunk=512)
        xi_epd = landy_szalay_xi(DD1, DR1, RR, nD=len(pos1), nR=len(posR))

        score = mse_on_bins(xi_epd, xi_obs, r_centers, rmax=60.0)

        if best is None or score &lt; best[0]:
            best = (score, sigma, lam, xi_epd)

best_score, best_sigma, best_lam, xi_best = best

# ==========================================
# 8) Plot baseline vs best EPD vs observed
# ==========================================
plt.figure()
plt.plot(r_centers, xi_base, marker='o', label=&quot;Mock baseline (uniform Poisson)&quot;)
plt.plot(r_centers, xi_best, marker='o',
         label=f&quot;Best EPD-resampled (sigma={best_sigma}, lambda={best_lam})&quot;)
plt.plot(r_centers, xi_obs, marker='o', linestyle='--',
         label=&quot;Observed BOSS DR11 ξ₀(r) (reference)&quot;)
plt.axhline(0, linewidth=1)
plt.xlabel(&quot;r [Mpc/h]&quot;)
plt.ylabel(&quot;ξ(r) (monopole estimate)&quot;)
plt.title(&quot;EPD reweighting of mock catalogs: mini-grid tuning (toy demonstration)&quot;)
plt.legend()
plt.show()

print(&quot;=== Mini-grid search result ===&quot;)
print(f&quot;Best score (MSE on r&lt;=60): {best_score:.6e}&quot;)
print(f&quot;Best sigma: {best_sigma}  Best lambda: {best_lam}&quot;)
&lt;/syntaxhighlight&gt;

'''How to run:''' download the BOSS multipoles text file to your computer and set &lt;code&gt;path&lt;/code&gt; in the script to that local filename. Then run the script in Python (NumPy + Matplotlib required). The baseline curve should lie near &lt;math&gt;\xi\approx 0&lt;/math&gt;, while the EPD-resampled mock typically shows positive small-scale clustering.

= Scope and limitations =
EPD is a mathematical framework for measure transformations. It does not claim:
* a replacement theory for GR/QM,
* empirical confirmation without explicit predictions and tests.

== 8. Speculative Extensions and Geometric Renormalization ==
''The following section is exploratory and speculative. It proposes possible extensions of Einstein Probability Dilation (EPD) into probabilistic geometry, curvature-weighted path integrals, and geometric renormalization. These ideas are heuristic and are not presently established as formal derivations or experimentally verified physical models.''

The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric principles into relativistic and quantum contexts.

Recent mathematical work extending Buffon’s needle problem to curved manifolds has suggested a possible connection between probability distributions and intrinsic geometry. In particular, studies of “Buffon deficits” on Gaussian manifolds demonstrated that deviations from the classical flat-space Buffon probability may encode Gaussian curvature through second-order geometric corrections.

In the classical flat Buffon problem, the expected crossing probability approaches:

[

\frac{2}{\pi}

]

for sufficiently small needles. On curved manifolds, however, the crossing probability acquires curvature-dependent corrections. In the small-length limit, these deviations may be interpreted as probabilistic signatures of intrinsic geometry.

This observation motivates a speculative extension of Einstein Probability Dilation (EPD): spacetime curvature may not only influence probability distributions, but probability structure itself may participate in the emergence or modulation of geometric behavior.

Within this interpretation, EPD may be viewed as a probabilistic-geometric framework in which curvature acts as a statistical weighting mechanism on path behavior. Rather than treating quantum path integrals strictly as sums over equally admissible histories, the geometry of spacetime could modulate the statistical contribution of classes of paths through curvature-dependent probability measures.

A possible schematic form is:

[

\int \mathcal{D}[x];W(K);e^{iS[x]/\hbar},

]

where:

* (e^{iS[x]/\hbar}) represents the standard quantum probability amplitude,
* (W(K)) represents a curvature-dependent geometric weighting factor,
* (K) denotes local Gaussian or spacetime curvature.

Under this interpretation, the two probabilistic structures are not identical in mathematical character.

The Feynman term:

[

e^{iS/\hbar}

]

is a complex probability amplitude capable of quantum interference and phase cancellation, whereas the curvature-weighting term behaves more like a real geometric measure density acting on the statistical structure of path space itself.

Accordingly, EPD may not “mix” probabilities directly, but instead combine:

* a quantum interference structure,
* with a curvature-dependent probabilistic geometry.

The Buffon deficit framework suggests that curvature may appear as a probabilistic deviation from flat geometric behavior. In the EPD interpretation, this motivates the possibility that curvature-weighted probability measures could alter the statistical admissibility of quantum paths at extremely small scales.

If such curvature-weighting naturally suppresses pathological ultraviolet path contributions, the framework could potentially resemble a form of geometric renormalization in which divergence control emerges from geometric probability structure rather than purely algebraic subtraction procedures.

A speculative expansion of the geometric weighting term might take the form:

[

W(K,L) \approx 1 + cKL^2 + O(L^3),

]

where:

* (L) represents a characteristic microscopic path scale,
* (K) represents local curvature,
* (c) is a proportionality constant.

Such a correction would recover ordinary flat-space behavior in the limit:

[

K \rightarrow 0.

]

At present these ideas remain heuristic and conceptual. However, they suggest a possible bridge between:

* stochastic geometry,
* probabilistic curvature,
* quantum path integrals,
* and emergent spacetime structure.

=== Open Questions ===

* Can curvature-weighted path measures be formally derived from probabilistic geometry?
* Can Buffon deficit relations be generalized from Gaussian manifolds to relativistic spacetime manifolds?
* Does curvature-weighting naturally regulate ultraviolet divergences in quantum field theory?
* What is the relationship between Gaussian curvature corrections and spacetime curvature tensors in General Relativity?
* Could probabilistic-geometric weighting contribute to emergent spacetime models or stochastic gravity frameworks?

= 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.

= See also =
* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Cosmological principle|Cosmological principle]]
* [[w:Holographic principle|Holographic principle]]
* [[w:Fractal geometry|Fractal geometry]]
* [[w:Quantum field theory|Quantum field theory]]
* [[w:Langlands program|Langlands program]]

= References =
==== Probability / measure theory ====
* [[w:Measure (mathematics)|Measure (mathematics)]]
* [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]]
* [[w:Probability theory|Probability theory]]

==== Computing / simulation ====
* [[w:Monte Carlo method|Monte Carlo method]]
* [[w:Importance sampling|Importance sampling]]

== Copyright and licensing ==
This Wikiversity page is authored by Howard Richardson. Please attribute appropriately when reusing or adapting this material.

Earlier and related versions may also be available on OSF and arXiv under the licenses stated on those platforms. Where multiple versions exist, the license shown on the specific platform/version governs reuse of that version.</text>
      <sha1>23ap38gc67tqnn0n9s6eapgtzq1hg9j</sha1>
    </revision>
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      <id>2810520</id>
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        <id>2995240</id>
      </contributor>
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      <text bytes="23778" sha1="ibdn8z3ire6azhaplaqwpu6fa6ih1ss" xml:space="preserve">{{Research project}}
{{To be peer reviewed}}

= Einstein Probability Dilation (EPD) =

== Research abstract ==
'''Einstein Probability Dilation (EPD)''' 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.

EPD 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).

EPD 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 ==
EPD 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]]). EPD 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.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant: &lt;math&gt;Z(P,D)=\int_\Omega D\,dP&lt;/math&gt;.
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure, &lt;math&gt;\mathbb{E}_P[f]=\int_\Omega f\,dP&lt;/math&gt;.

== Probability dilation (reweighting) ==
Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D\,dP}{\int_\Omega D\,dP}\quad\text{for all }A\in\Sigma.&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt; where

&lt;math&gt;\widetilde{p}(x)=\frac{D(x)\,p(x)}{Z},\quad Z=\int_\Omega D(x)\,p(x)\,d\mu.&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

== Example: Lorentz contraction as a positive dilation field ==
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;\gamma(v)=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\quad\text{for } |v| &lt; c.&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;L(v)=\frac{L_0}{\gamma(v)}.&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;, for instance:
* &lt;math&gt;D(x)=\gamma(x)&lt;/math&gt; (weights states by increasing &lt;math&gt;\gamma&lt;/math&gt;), or
* &lt;math&gt;D(x)=\frac{1}{\gamma(x)}&lt;/math&gt; (weights states by decreasing &lt;math&gt;\gamma&lt;/math&gt;),

provided &lt;math&gt;D(x)&lt;/math&gt; is strictly positive over the model’s domain and &lt;math&gt;0&lt;\int_\Omega D\,dP&lt;\infty&lt;/math&gt;.

This example is included purely to illustrate “positive dilation”; it is not presented as a cosmological model.

=== Buffon sampling, the constant \pi, and a Lorentz-induced dilation factor (toy) ===
This brief example shows how &lt;math&gt;\pi&lt;/math&gt; naturally appears from sampling geometry (Buffon’s needle) and how inserting Lorentz contraction into the sampled length produces a strictly positive, dimensionless candidate dilation field. This is included for mathematical intuition; it is not claimed as a physical derivation.

For the classical [[w:Buffon's needle problem|Buffon needle problem]] with needle length &lt;math&gt;L\le d&lt;/math&gt; and line spacing &lt;math&gt;d&lt;/math&gt;, the crossing probability is:
&lt;math&gt;P_{\mathrm{cross}}(L;d)=\frac{2L}{\pi d}.&lt;/math&gt;

If a rest-length &lt;math&gt;L_0&lt;/math&gt; undergoes Lorentz contraction &lt;math&gt;L(v)=L_0/\gamma(v)&lt;/math&gt; for &lt;math&gt;|v| &lt; c&lt;/math&gt;, then the same form gives:
&lt;math&gt;P_{\mathrm{cross}}(v)=\frac{2L_0}{\pi d\,\gamma(v)}.&lt;/math&gt;

Normalizing to the rest case yields a dimensionless sampling ratio:
&lt;math&gt;D_{\pi}(v):=\frac{P_{\mathrm{cross}}(v)}{P_{\mathrm{cross}}(0)}=\frac{1}{\gamma(v)}=\sqrt{1-\frac{v^2}{c^2}}.&lt;/math&gt;

This &lt;math&gt;D_{\pi}(v)&lt;/math&gt; is strictly positive on the domain &lt;math&gt;|v| &lt; c&lt;/math&gt; and can be used (illustratively) as a dilation field in the EPD transform:
&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D_{\pi}(v)\,dP}{\int_\Omega D_{\pi}(v)\,dP}.&lt;/math&gt;

== Theorem-style mathematics expansion ==

=== Definition (EPD reweighting operator) ===
Let &lt;math&gt;P&lt;/math&gt; be a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; and &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; measurable with &lt;math&gt;0&lt;\int_\Omega D\,dP&lt;\infty&lt;/math&gt;. Define &lt;math&gt;\mathrm{EPD}(P;D)=\widetilde{P}&lt;/math&gt; by:

&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D\,dP}{\int_\Omega D\,dP}.&lt;/math&gt;

=== Theorem 1 (Normalization and positivity) ===
&lt;math&gt;\widetilde{P}&lt;/math&gt; is a probability measure. In particular:
# &lt;math&gt;\widetilde{P}(A)\ge 0&lt;/math&gt; for all measurable &lt;math&gt;A&lt;/math&gt;,
# &lt;math&gt;\widetilde{P}(\Omega)=1&lt;/math&gt;,
# If &lt;math&gt;P(A)=0&lt;/math&gt; then &lt;math&gt;\widetilde{P}(A)=0&lt;/math&gt; (absolute continuity preserved).

=== Theorem 2 (Expectation reweighting identity) ===
If &lt;math&gt;f&lt;/math&gt; is integrable under &lt;math&gt;\widetilde{P}&lt;/math&gt;, then

&lt;math&gt;\mathbb{E}_{\widetilde{P}}[f]=\frac{\mathbb{E}_P[D\,f]}{\mathbb{E}_P[D]}.&lt;/math&gt;

=== Theorem 3 (Composition / iteration rule) ===
Let &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D_1)&lt;/math&gt; and &lt;math&gt;\widehat{P}=\mathrm{EPD}(\widetilde{P};D_2)&lt;/math&gt;. Then

&lt;math&gt;\widehat{P}=\mathrm{EPD}(P;D_1D_2)&lt;/math&gt;

provided &lt;math&gt;\int_\Omega D_1D_2\,dP&lt;/math&gt; is finite and nonzero.

=== Theorem 4 (Fixed-point condition) ===
&lt;math&gt;P&lt;/math&gt; is a fixed point of EPD under dilation &lt;math&gt;D&lt;/math&gt; (i.e., &lt;math&gt;\mathrm{EPD}(P;D)=P&lt;/math&gt;) if and only if &lt;math&gt;D&lt;/math&gt; is constant &lt;math&gt;P&lt;/math&gt;-almost everywhere.

== Invariant quantities ==
EPD naturally highlights quantities that are invariant (or transform in controlled ways) under reweighting, including:
* ratios of expectations of the form &lt;math&gt;\mathbb{E}_P[D f]/\mathbb{E}_P[D]&lt;/math&gt;,
* equivalence classes of observables that remain unchanged under specific families of dilations,
* fixed-point structure (trivial dilations) and near-fixed behavior (small perturbations).

= Core axioms of Einstein Probability Dilation =

=== Axiom 1 — Probability primacy ===
Probability measures are treated as primary mathematical objects; structure is inferred from invariant properties of measure transformations.

=== Axiom 2 — Positive dilation ===
Dilation fields are positive (&lt;math&gt;D&lt;/math&gt; is strictly positive) so that reweighting preserves positivity and supports a consistent normalization constant.

=== Axiom 3 — Iterative composability ===
Successive dilations compose multiplicatively (&lt;math&gt;D_1D_2&lt;/math&gt;), enabling multi-step (iterative) dynamics on measures.

= Falsifiability conditions =
A proposed EPD-based model should specify:
* the configuration space &lt;math&gt;\Omega&lt;/math&gt;,
* a baseline measure &lt;math&gt;P&lt;/math&gt;,
* an explicit dilation field &lt;math&gt;D&lt;/math&gt; (with parameters and domain),
* measurable predictions (statistics, scaling laws, invariants) that can be checked against simulation or data,
* a baseline comparison (null model) and evaluation criteria (error measures, goodness-of-fit, or hypothesis tests).

'''Concrete falsifiability test (clustering example):''' Choose a baseline mock galaxy catalog (configuration space &lt;math&gt;\Omega&lt;/math&gt;), define an explicit strictly positive dilation field &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt;, apply EPD-style reweighting (e.g., importance resampling of the empirical measure), and compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt; using a standard estimator (e.g., normalized Landy–Szalay). Define a quantitative success metric (for example, mean-squared error between modeled and observed &lt;math&gt;\xi(r)&lt;/math&gt; over specified separation bins such as &lt;math&gt;r\le 60\,\mathrm{Mpc}/h&lt;/math&gt;). The model is falsified if no parameter choices within the stated domain can meet the pre-registered threshold under the chosen metric, or if the fit fails out-of-sample on withheld bins.

= Numerical simulation and iterative models =

== Simulation model description ==
In discrete demonstrations, the “state space” may be a finite set (e.g., bins, configurations, or catalog points), and the probability measure &lt;math&gt;P&lt;/math&gt; may be represented as a probability vector (or an empirical measure) over those states.

Two equivalent discrete implementations are common:
* '''weighted evaluation''': keep all points and carry weights proportional to &lt;math&gt;D&lt;/math&gt;, or
* '''importance resampling''': draw a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs and comparing \xi(r) ===
This demonstration connects the EPD reweighting idea to a standard observable in large-scale structure: the (monopole) two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;. The goal here is not a precision cosmology fit, but a clean, falsifiable pipeline:

# generate a baseline mock galaxy catalog in a periodic box (approximately unclustered),
# define a strictly positive dilation field &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt; over galaxy positions,
# perform EPD-style importance resampling to obtain a reweighted (clustered) mock catalog,
# measure &lt;math&gt;\xi(r)&lt;/math&gt; using the properly normalized Landy–Szalay estimator,
# tune a small mini-grid over &lt;math&gt;(\sigma,\lambda)&lt;/math&gt; to minimize a simple mismatch score on small scales.

==== Key definitions ====
* Dilation field: &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt;, with &lt;math&gt;\lambda &gt; 0&lt;/math&gt; and &lt;math&gt;\phi(x)\ge 0&lt;/math&gt;.
* Example potential: &lt;math&gt;\phi(x)=\sum_k \exp\!\left(-\|x-s_k\|^2/(2\sigma^2)\right)&lt;/math&gt; using seed points &lt;math&gt;s_k&lt;/math&gt;.
* Landy–Szalay estimator (normalized counts):
  &lt;math&gt;\xi(r)=\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}&lt;/math&gt;,
  where &lt;math&gt;DD,DR,RR&lt;/math&gt; are pair counts normalized by the total number of possible pairs in each catalog.

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
import numpy as np
import matplotlib.pyplot as plt

# ==========================================
# 1) Pair-counting with periodic boundaries
# ==========================================
def pair_counts_weighted(posA, posB, bins, boxsize, wA=None, wB=None,
                         autocorr=False, chunk=512):
    &quot;&quot;&quot;
    Weighted pair counts in radial bins with periodic boundary conditions.
    - autocorr=True: count unique pairs i&lt;j within posA
    - autocorr=False: count all cross pairs between posA and posB
    Returns raw (un-normalized) weighted counts per bin.
    &quot;&quot;&quot;
    posA = np.asarray(posA, float)
    posB = np.asarray(posB, float)
    bins = np.asarray(bins, float)
    nb = len(bins) - 1
    L = float(boxsize)

    if wA is None:
        wA = np.ones(len(posA), float)
    if wB is None:
        wB = np.ones(len(posB), float)
    wA = np.asarray(wA, float)
    wB = np.asarray(wB, float)

    counts = np.zeros(nb, float)

    if autocorr:
        N = len(posA)
        for i0 in range(0, N, chunk):
            i1 = min(N, i0 + chunk)
            A = posA[i0:i1]
            wAi = wA[i0:i1]

            # distances from A to ALL posA (periodic)
            d = A[:, None, :] - posA[None, :, :]
            d -= L * np.round(d / L)
            r = np.sqrt(np.sum(d * d, axis=2))

            W = wAi[:, None] * wA[None, :]

            # accumulate only i&lt;j
            for ii in range(i1 - i0):
                j_start = i0 + ii + 1
                if j_start &gt;= N:
                    continue
                rr = r[ii, j_start:]
                ww = W[ii, j_start:]
                hist, _ = np.histogram(rr, bins=bins, weights=ww)
                counts += hist
        return counts

    else:
        NA = len(posA)
        for i0 in range(0, NA, chunk):
            i1 = min(NA, i0 + chunk)
            A = posA[i0:i1]
            wAi = wA[i0:i1]

            d = A[:, None, :] - posB[None, :, :]
            d -= L * np.round(d / L)
            r = np.sqrt(np.sum(d * d, axis=2))

            W = wAi[:, None] * wB[None, :]
            hist, _ = np.histogram(r, bins=bins, weights=W)
            counts += hist
        return counts


def landy_szalay_xi(DD_raw, DR_raw, RR_raw, nD, nR):
    &quot;&quot;&quot;
    Properly normalized Landy–Szalay estimator:
      DD = DD_raw / [nD (nD-1)/2]
      DR = DR_raw / [nD nR]
      RR = RR_raw / [nR (nR-1)/2]
      xi = (DD - 2 DR + RR) / RR
    &quot;&quot;&quot;
    DD = DD_raw / (nD * (nD - 1) / 2.0)
    DR = DR_raw / (nD * nR)
    RR = RR_raw / (nR * (nR - 1) / 2.0)
    return (DD - 2 * DR + RR) / np.maximum(RR, 1e-30)


# ==========================================
# 2) EPD-style dilation field + resampling
# ==========================================
def gaussian_seed_field(pos, seeds, boxsize, sigma):
    &quot;&quot;&quot;
    phi(x) = sum_k exp(-||x-seed_k||^2/(2 sigma^2)) with periodic distances.
    Always nonnegative.
    &quot;&quot;&quot;
    pos = np.asarray(pos, float)
    seeds = np.asarray(seeds, float)
    L = float(boxsize)

    d = pos[:, None, :] - seeds[None, :, :]
    d -= L * np.round(d / L)
    r2 = np.sum(d * d, axis=2)
    phi = np.exp(-0.5 * r2 / (sigma ** 2)).sum(axis=1)
    return phi


def epd_resample_catalog(pos0, phi, lam, rng):
    &quot;&quot;&quot;
    Importance-resampling version of discrete EPD on an empirical measure:
      weights ∝ D(x)=exp(lam * phi(x))
      resample points with prob ∝ weights
    &quot;&quot;&quot;
    logw = lam * phi
    logw -= np.max(logw)          # stabilize
    w = np.exp(logw)              # strictly positive
    p = w / w.sum()
    idx = rng.choice(len(pos0), size=len(pos0), replace=True, p=p)
    return pos0[idx]


# ==========================================
# 3) Load real xi0(r) for reference (BOSS)
# ==========================================
# NOTE: Set this path to wherever you saved the file locally.
# The example below matches the earlier DR11 multipoles file name.
path = &quot;samushia_2013_CMASSDR11_xi_multipoles.dat&quot;
vals = np.loadtxt(path)
xi_obs = vals[:16]  # monopole xi0

# BOSS bins: edges 24..152 step 8 =&gt; centers 28..148
bins = np.arange(24, 152 + 8, 8)
r_centers = (bins[:-1] + bins[1:]) / 2


# ==========================================
# 4) Build baseline mock + randoms (periodic box)
# ==========================================
rng = np.random.default_rng(7)

boxsize = 200.0
N_data0 = 1400
N_rand = 4500

pos0 = rng.uniform(0, boxsize, size=(N_data0, 3))
posR = rng.uniform(0, boxsize, size=(N_rand, 3))

# Precompute RR once (random-random doesn't change during grid search)
RR = pair_counts_weighted(posR, posR, bins, boxsize, autocorr=True, chunk=512)


# ==========================================
# 5) Define seed locations (kept fixed during tuning)
# ==========================================
K = 12
seeds = rng.uniform(0, boxsize, size=(K, 3))


# ==========================================
# 6) Objective: simple &quot;success&quot; on small-r bins
# ==========================================
def mse_on_bins(xi_model, xi_target, r_centers, rmax=60.0):
    &quot;&quot;&quot;
    Mean squared error over bins with r &lt;= rmax (simple success criterion).
    &quot;&quot;&quot;
    m = r_centers &lt;= rmax
    return float(np.mean((xi_model[m] - xi_target[m]) ** 2))


# ==========================================
# 7) Mini-grid search over (lambda, sigma)
# ==========================================
sigmas = [6.0, 8.0, 10.0, 12.0, 15.0]   # seed width (correlation length scale)
lams   = [0.5, 1.0, 2.0, 3.0, 4.0, 5.0] # dilation strength

best = None

# Baseline xi (should be ~0)
DD0 = pair_counts_weighted(pos0, pos0, bins, boxsize, autocorr=True, chunk=512)
DR0 = pair_counts_weighted(pos0, posR, bins, boxsize, autocorr=False, chunk=512)
xi_base = landy_szalay_xi(DD0, DR0, RR, nD=len(pos0), nR=len(posR))

for sigma in sigmas:
    phi = gaussian_seed_field(pos0, seeds, boxsize, sigma)

    for lam in lams:
        pos1 = epd_resample_catalog(pos0, phi, lam, rng)

        DD1 = pair_counts_weighted(pos1, pos1, bins, boxsize, autocorr=True, chunk=512)
        DR1 = pair_counts_weighted(pos1, posR, bins, boxsize, autocorr=False, chunk=512)
        xi_epd = landy_szalay_xi(DD1, DR1, RR, nD=len(pos1), nR=len(posR))

        score = mse_on_bins(xi_epd, xi_obs, r_centers, rmax=60.0)

        if best is None or score &lt; best[0]:
            best = (score, sigma, lam, xi_epd)

best_score, best_sigma, best_lam, xi_best = best

# ==========================================
# 8) Plot baseline vs best EPD vs observed
# ==========================================
plt.figure()
plt.plot(r_centers, xi_base, marker='o', label=&quot;Mock baseline (uniform Poisson)&quot;)
plt.plot(r_centers, xi_best, marker='o',
         label=f&quot;Best EPD-resampled (sigma={best_sigma}, lambda={best_lam})&quot;)
plt.plot(r_centers, xi_obs, marker='o', linestyle='--',
         label=&quot;Observed BOSS DR11 ξ₀(r) (reference)&quot;)
plt.axhline(0, linewidth=1)
plt.xlabel(&quot;r [Mpc/h]&quot;)
plt.ylabel(&quot;ξ(r) (monopole estimate)&quot;)
plt.title(&quot;EPD reweighting of mock catalogs: mini-grid tuning (toy demonstration)&quot;)
plt.legend()
plt.show()

print(&quot;=== Mini-grid search result ===&quot;)
print(f&quot;Best score (MSE on r&lt;=60): {best_score:.6e}&quot;)
print(f&quot;Best sigma: {best_sigma}  Best lambda: {best_lam}&quot;)
&lt;/syntaxhighlight&gt;

'''How to run:''' download the BOSS multipoles text file to your computer and set &lt;code&gt;path&lt;/code&gt; in the script to that local filename. Then run the script in Python (NumPy + Matplotlib required). The baseline curve should lie near &lt;math&gt;\xi\approx 0&lt;/math&gt;, while the EPD-resampled mock typically shows positive small-scale clustering.

= Scope and limitations =
EPD is a mathematical framework for measure transformations. It does not claim:
* a replacement theory for GR/QM,
* empirical confirmation without explicit predictions and tests.

== 8. Speculative Extensions and Geometric Renormalization ==
== Speculative Extensions and Geometric Renormalization ==

{{Disclaimer|This section is speculative and exploratory in nature.}}

Recent mathematical work extending Buffon's needle problem to curved manifolds suggests a possible connection between probability distributions and intrinsic geometry. In particular, studies of &quot;Buffon deficits&quot; on Gaussian manifolds indicate that deviations from the classical flat-space Buffon probability may encode Gaussian curvature through small geometric corrections.

In flat geometry, the classical Buffon crossing probability approaches:

&lt;math&gt;
\frac{2}{\pi}
&lt;/math&gt;

for sufficiently small needles.

On curved manifolds, however, the crossing probability acquires curvature-dependent corrections. This motivates a speculative extension of Einstein Probability Dilation (EPD): spacetime curvature may not only influence probability distributions, but probability structure itself may participate in the emergence or modulation of geometric behavior.

Within this interpretation, curvature may act as a statistical weighting mechanism on quantum path behavior. Rather than treating quantum path integrals strictly as sums over equally admissible histories, spacetime geometry could modulate the statistical contribution of classes of paths through curvature-dependent weighting functions.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x \, W(K)\, e^{iS/\hbar}
&lt;/math&gt;

where:

&lt;math&gt;e^{iS/\hbar}&lt;/math&gt;

represents the standard quantum probability amplitude,

&lt;math&gt;W(K)&lt;/math&gt;

represents a curvature-dependent geometric weighting factor,

and

&lt;math&gt;K&lt;/math&gt;

represents local Gaussian or spacetime curvature.

In this interpretation, the two probabilistic structures are not identical in mathematical character.

The Feynman factor

&lt;math&gt;e^{iS/\hbar}&lt;/math&gt;

is a complex quantum probability amplitude capable of interference and phase cancellation, whereas the curvature-weighting term behaves more like a real geometric measure acting on the statistical structure of path space itself.

Accordingly, EPD may not combine probabilities directly, but instead combine:
* a quantum interference structure,
* with a curvature-dependent probabilistic geometry.

The Buffon deficit framework suggests that curvature may appear as a probabilistic deviation from flat geometric behavior. Within EPD, this motivates the possibility that curvature-weighted probability measures could alter the statistical admissibility of quantum paths at extremely small scales.

If such curvature-weighting naturally suppresses pathological ultraviolet path contributions, the framework could potentially resemble a form of geometric renormalization in which divergence control emerges from geometric probability structure rather than purely algebraic subtraction procedures.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)\approx 1+cKL^2+O(L^3)
&lt;/math&gt;

where:

&lt;math&gt;L&lt;/math&gt;

represents a microscopic path scale,

&lt;math&gt;K&lt;/math&gt;

represents local curvature,

and

&lt;math&gt;c&lt;/math&gt;

represents a proportionality constant.

In the flat-space limit:

&lt;math&gt;
K\rightarrow0
&lt;/math&gt;

ordinary quantum behavior would be recovered.

At present these ideas remain heuristic and conceptual. However, they suggest a possible bridge between:
* stochastic geometry,
* probabilistic curvature,
* quantum path integrals,
* and emergent spacetime structure.

=== Open Questions ===

* Can curvature-weighted path measures be formally derived from probabilistic geometry?
* Can Buffon deficit relations be generalized from Gaussian manifolds to relativistic spacetime manifolds?
* Does curvature-weighting naturally regulate ultraviolet divergences in quantum field theory?
* What is the relationship between Gaussian curvature corrections and spacetime curvature tensors in General Relativity?
* Could probabilistic-geometric weighting contribute to emergent spacetime or stochastic gravity models?

= 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.

= See also =
* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Cosmological principle|Cosmological principle]]
* [[w:Holographic principle|Holographic principle]]
* [[w:Fractal geometry|Fractal geometry]]
* [[w:Quantum field theory|Quantum field theory]]
* [[w:Langlands program|Langlands program]]

= References =
==== Probability / measure theory ====
* [[w:Measure (mathematics)|Measure (mathematics)]]
* [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]]
* [[w:Probability theory|Probability theory]]

==== Computing / simulation ====
* [[w:Monte Carlo method|Monte Carlo method]]
* [[w:Importance sampling|Importance sampling]]

== Copyright and licensing ==
This Wikiversity page is authored by Howard Richardson. Please attribute appropriately when reusing or adapting this material.

Earlier and related versions may also be available on OSF and arXiv under the licenses stated on those platforms. Where multiple versions exist, the license shown on the specific platform/version governs reuse of that version.</text>
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      <text bytes="24219" sha1="n7s0ikkkwr30k75itfc9jc6af4uv3z5" xml:space="preserve">{{Research project}}
{{To be peer reviewed}}

= Einstein Probability Dilation (EPD) =

== Research abstract ==
'''Einstein Probability Dilation (EPD)''' 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.

EPD 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).

EPD 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 ==
EPD 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]]). EPD 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.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant: &lt;math&gt;Z(P,D)=\int_\Omega D\,dP&lt;/math&gt;.
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure, &lt;math&gt;\mathbb{E}_P[f]=\int_\Omega f\,dP&lt;/math&gt;.

== Probability dilation (reweighting) ==
Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D\,dP}{\int_\Omega D\,dP}\quad\text{for all }A\in\Sigma.&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt; where

&lt;math&gt;\widetilde{p}(x)=\frac{D(x)\,p(x)}{Z},\quad Z=\int_\Omega D(x)\,p(x)\,d\mu.&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

== Example: Lorentz contraction as a positive dilation field ==
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;\gamma(v)=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\quad\text{for } |v| &lt; c.&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;L(v)=\frac{L_0}{\gamma(v)}.&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;, for instance:
* &lt;math&gt;D(x)=\gamma(x)&lt;/math&gt; (weights states by increasing &lt;math&gt;\gamma&lt;/math&gt;), or
* &lt;math&gt;D(x)=\frac{1}{\gamma(x)}&lt;/math&gt; (weights states by decreasing &lt;math&gt;\gamma&lt;/math&gt;),

provided &lt;math&gt;D(x)&lt;/math&gt; is strictly positive over the model’s domain and &lt;math&gt;0&lt;\int_\Omega D\,dP&lt;\infty&lt;/math&gt;.

This example is included purely to illustrate “positive dilation”; it is not presented as a cosmological model.

=== Buffon sampling, the constant \pi, and a Lorentz-induced dilation factor (toy) ===
This brief example shows how &lt;math&gt;\pi&lt;/math&gt; naturally appears from sampling geometry (Buffon’s needle) and how inserting Lorentz contraction into the sampled length produces a strictly positive, dimensionless candidate dilation field. This is included for mathematical intuition; it is not claimed as a physical derivation.

For the classical [[w:Buffon's needle problem|Buffon needle problem]] with needle length &lt;math&gt;L\le d&lt;/math&gt; and line spacing &lt;math&gt;d&lt;/math&gt;, the crossing probability is:
&lt;math&gt;P_{\mathrm{cross}}(L;d)=\frac{2L}{\pi d}.&lt;/math&gt;

If a rest-length &lt;math&gt;L_0&lt;/math&gt; undergoes Lorentz contraction &lt;math&gt;L(v)=L_0/\gamma(v)&lt;/math&gt; for &lt;math&gt;|v| &lt; c&lt;/math&gt;, then the same form gives:
&lt;math&gt;P_{\mathrm{cross}}(v)=\frac{2L_0}{\pi d\,\gamma(v)}.&lt;/math&gt;

Normalizing to the rest case yields a dimensionless sampling ratio:
&lt;math&gt;D_{\pi}(v):=\frac{P_{\mathrm{cross}}(v)}{P_{\mathrm{cross}}(0)}=\frac{1}{\gamma(v)}=\sqrt{1-\frac{v^2}{c^2}}.&lt;/math&gt;

This &lt;math&gt;D_{\pi}(v)&lt;/math&gt; is strictly positive on the domain &lt;math&gt;|v| &lt; c&lt;/math&gt; and can be used (illustratively) as a dilation field in the EPD transform:
&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D_{\pi}(v)\,dP}{\int_\Omega D_{\pi}(v)\,dP}.&lt;/math&gt;

== Theorem-style mathematics expansion ==

=== Definition (EPD reweighting operator) ===
Let &lt;math&gt;P&lt;/math&gt; be a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; and &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; measurable with &lt;math&gt;0&lt;\int_\Omega D\,dP&lt;\infty&lt;/math&gt;. Define &lt;math&gt;\mathrm{EPD}(P;D)=\widetilde{P}&lt;/math&gt; by:

&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D\,dP}{\int_\Omega D\,dP}.&lt;/math&gt;

=== Theorem 1 (Normalization and positivity) ===
&lt;math&gt;\widetilde{P}&lt;/math&gt; is a probability measure. In particular:
# &lt;math&gt;\widetilde{P}(A)\ge 0&lt;/math&gt; for all measurable &lt;math&gt;A&lt;/math&gt;,
# &lt;math&gt;\widetilde{P}(\Omega)=1&lt;/math&gt;,
# If &lt;math&gt;P(A)=0&lt;/math&gt; then &lt;math&gt;\widetilde{P}(A)=0&lt;/math&gt; (absolute continuity preserved).

=== Theorem 2 (Expectation reweighting identity) ===
If &lt;math&gt;f&lt;/math&gt; is integrable under &lt;math&gt;\widetilde{P}&lt;/math&gt;, then

&lt;math&gt;\mathbb{E}_{\widetilde{P}}[f]=\frac{\mathbb{E}_P[D\,f]}{\mathbb{E}_P[D]}.&lt;/math&gt;

=== Theorem 3 (Composition / iteration rule) ===
Let &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D_1)&lt;/math&gt; and &lt;math&gt;\widehat{P}=\mathrm{EPD}(\widetilde{P};D_2)&lt;/math&gt;. Then

&lt;math&gt;\widehat{P}=\mathrm{EPD}(P;D_1D_2)&lt;/math&gt;

provided &lt;math&gt;\int_\Omega D_1D_2\,dP&lt;/math&gt; is finite and nonzero.

=== Theorem 4 (Fixed-point condition) ===
&lt;math&gt;P&lt;/math&gt; is a fixed point of EPD under dilation &lt;math&gt;D&lt;/math&gt; (i.e., &lt;math&gt;\mathrm{EPD}(P;D)=P&lt;/math&gt;) if and only if &lt;math&gt;D&lt;/math&gt; is constant &lt;math&gt;P&lt;/math&gt;-almost everywhere.

== Invariant quantities ==
EPD naturally highlights quantities that are invariant (or transform in controlled ways) under reweighting, including:
* ratios of expectations of the form &lt;math&gt;\mathbb{E}_P[D f]/\mathbb{E}_P[D]&lt;/math&gt;,
* equivalence classes of observables that remain unchanged under specific families of dilations,
* fixed-point structure (trivial dilations) and near-fixed behavior (small perturbations).

= Core axioms of Einstein Probability Dilation =

=== Axiom 1 — Probability primacy ===
Probability measures are treated as primary mathematical objects; structure is inferred from invariant properties of measure transformations.

=== Axiom 2 — Positive dilation ===
Dilation fields are positive (&lt;math&gt;D&lt;/math&gt; is strictly positive) so that reweighting preserves positivity and supports a consistent normalization constant.

=== Axiom 3 — Iterative composability ===
Successive dilations compose multiplicatively (&lt;math&gt;D_1D_2&lt;/math&gt;), enabling multi-step (iterative) dynamics on measures.

= Falsifiability conditions =
A proposed EPD-based model should specify:
* the configuration space &lt;math&gt;\Omega&lt;/math&gt;,
* a baseline measure &lt;math&gt;P&lt;/math&gt;,
* an explicit dilation field &lt;math&gt;D&lt;/math&gt; (with parameters and domain),
* measurable predictions (statistics, scaling laws, invariants) that can be checked against simulation or data,
* a baseline comparison (null model) and evaluation criteria (error measures, goodness-of-fit, or hypothesis tests).

'''Concrete falsifiability test (clustering example):''' Choose a baseline mock galaxy catalog (configuration space &lt;math&gt;\Omega&lt;/math&gt;), define an explicit strictly positive dilation field &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt;, apply EPD-style reweighting (e.g., importance resampling of the empirical measure), and compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt; using a standard estimator (e.g., normalized Landy–Szalay). Define a quantitative success metric (for example, mean-squared error between modeled and observed &lt;math&gt;\xi(r)&lt;/math&gt; over specified separation bins such as &lt;math&gt;r\le 60\,\mathrm{Mpc}/h&lt;/math&gt;). The model is falsified if no parameter choices within the stated domain can meet the pre-registered threshold under the chosen metric, or if the fit fails out-of-sample on withheld bins.

= Numerical simulation and iterative models =

== Simulation model description ==
In discrete demonstrations, the “state space” may be a finite set (e.g., bins, configurations, or catalog points), and the probability measure &lt;math&gt;P&lt;/math&gt; may be represented as a probability vector (or an empirical measure) over those states.

Two equivalent discrete implementations are common:
* '''weighted evaluation''': keep all points and carry weights proportional to &lt;math&gt;D&lt;/math&gt;, or
* '''importance resampling''': draw a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs and comparing \xi(r) ===
This demonstration connects the EPD reweighting idea to a standard observable in large-scale structure: the (monopole) two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;. The goal here is not a precision cosmology fit, but a clean, falsifiable pipeline:

# generate a baseline mock galaxy catalog in a periodic box (approximately unclustered),
# define a strictly positive dilation field &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt; over galaxy positions,
# perform EPD-style importance resampling to obtain a reweighted (clustered) mock catalog,
# measure &lt;math&gt;\xi(r)&lt;/math&gt; using the properly normalized Landy–Szalay estimator,
# tune a small mini-grid over &lt;math&gt;(\sigma,\lambda)&lt;/math&gt; to minimize a simple mismatch score on small scales.

==== Key definitions ====
* Dilation field: &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt;, with &lt;math&gt;\lambda &gt; 0&lt;/math&gt; and &lt;math&gt;\phi(x)\ge 0&lt;/math&gt;.
* Example potential: &lt;math&gt;\phi(x)=\sum_k \exp\!\left(-\|x-s_k\|^2/(2\sigma^2)\right)&lt;/math&gt; using seed points &lt;math&gt;s_k&lt;/math&gt;.
* Landy–Szalay estimator (normalized counts):
  &lt;math&gt;\xi(r)=\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}&lt;/math&gt;,
  where &lt;math&gt;DD,DR,RR&lt;/math&gt; are pair counts normalized by the total number of possible pairs in each catalog.

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
import numpy as np
import matplotlib.pyplot as plt

# ==========================================
# 1) Pair-counting with periodic boundaries
# ==========================================
def pair_counts_weighted(posA, posB, bins, boxsize, wA=None, wB=None,
                         autocorr=False, chunk=512):
    &quot;&quot;&quot;
    Weighted pair counts in radial bins with periodic boundary conditions.
    - autocorr=True: count unique pairs i&lt;j within posA
    - autocorr=False: count all cross pairs between posA and posB
    Returns raw (un-normalized) weighted counts per bin.
    &quot;&quot;&quot;
    posA = np.asarray(posA, float)
    posB = np.asarray(posB, float)
    bins = np.asarray(bins, float)
    nb = len(bins) - 1
    L = float(boxsize)

    if wA is None:
        wA = np.ones(len(posA), float)
    if wB is None:
        wB = np.ones(len(posB), float)
    wA = np.asarray(wA, float)
    wB = np.asarray(wB, float)

    counts = np.zeros(nb, float)

    if autocorr:
        N = len(posA)
        for i0 in range(0, N, chunk):
            i1 = min(N, i0 + chunk)
            A = posA[i0:i1]
            wAi = wA[i0:i1]

            # distances from A to ALL posA (periodic)
            d = A[:, None, :] - posA[None, :, :]
            d -= L * np.round(d / L)
            r = np.sqrt(np.sum(d * d, axis=2))

            W = wAi[:, None] * wA[None, :]

            # accumulate only i&lt;j
            for ii in range(i1 - i0):
                j_start = i0 + ii + 1
                if j_start &gt;= N:
                    continue
                rr = r[ii, j_start:]
                ww = W[ii, j_start:]
                hist, _ = np.histogram(rr, bins=bins, weights=ww)
                counts += hist
        return counts

    else:
        NA = len(posA)
        for i0 in range(0, NA, chunk):
            i1 = min(NA, i0 + chunk)
            A = posA[i0:i1]
            wAi = wA[i0:i1]

            d = A[:, None, :] - posB[None, :, :]
            d -= L * np.round(d / L)
            r = np.sqrt(np.sum(d * d, axis=2))

            W = wAi[:, None] * wB[None, :]
            hist, _ = np.histogram(r, bins=bins, weights=W)
            counts += hist
        return counts


def landy_szalay_xi(DD_raw, DR_raw, RR_raw, nD, nR):
    &quot;&quot;&quot;
    Properly normalized Landy–Szalay estimator:
      DD = DD_raw / [nD (nD-1)/2]
      DR = DR_raw / [nD nR]
      RR = RR_raw / [nR (nR-1)/2]
      xi = (DD - 2 DR + RR) / RR
    &quot;&quot;&quot;
    DD = DD_raw / (nD * (nD - 1) / 2.0)
    DR = DR_raw / (nD * nR)
    RR = RR_raw / (nR * (nR - 1) / 2.0)
    return (DD - 2 * DR + RR) / np.maximum(RR, 1e-30)


# ==========================================
# 2) EPD-style dilation field + resampling
# ==========================================
def gaussian_seed_field(pos, seeds, boxsize, sigma):
    &quot;&quot;&quot;
    phi(x) = sum_k exp(-||x-seed_k||^2/(2 sigma^2)) with periodic distances.
    Always nonnegative.
    &quot;&quot;&quot;
    pos = np.asarray(pos, float)
    seeds = np.asarray(seeds, float)
    L = float(boxsize)

    d = pos[:, None, :] - seeds[None, :, :]
    d -= L * np.round(d / L)
    r2 = np.sum(d * d, axis=2)
    phi = np.exp(-0.5 * r2 / (sigma ** 2)).sum(axis=1)
    return phi


def epd_resample_catalog(pos0, phi, lam, rng):
    &quot;&quot;&quot;
    Importance-resampling version of discrete EPD on an empirical measure:
      weights ∝ D(x)=exp(lam * phi(x))
      resample points with prob ∝ weights
    &quot;&quot;&quot;
    logw = lam * phi
    logw -= np.max(logw)          # stabilize
    w = np.exp(logw)              # strictly positive
    p = w / w.sum()
    idx = rng.choice(len(pos0), size=len(pos0), replace=True, p=p)
    return pos0[idx]


# ==========================================
# 3) Load real xi0(r) for reference (BOSS)
# ==========================================
# NOTE: Set this path to wherever you saved the file locally.
# The example below matches the earlier DR11 multipoles file name.
path = &quot;samushia_2013_CMASSDR11_xi_multipoles.dat&quot;
vals = np.loadtxt(path)
xi_obs = vals[:16]  # monopole xi0

# BOSS bins: edges 24..152 step 8 =&gt; centers 28..148
bins = np.arange(24, 152 + 8, 8)
r_centers = (bins[:-1] + bins[1:]) / 2


# ==========================================
# 4) Build baseline mock + randoms (periodic box)
# ==========================================
rng = np.random.default_rng(7)

boxsize = 200.0
N_data0 = 1400
N_rand = 4500

pos0 = rng.uniform(0, boxsize, size=(N_data0, 3))
posR = rng.uniform(0, boxsize, size=(N_rand, 3))

# Precompute RR once (random-random doesn't change during grid search)
RR = pair_counts_weighted(posR, posR, bins, boxsize, autocorr=True, chunk=512)


# ==========================================
# 5) Define seed locations (kept fixed during tuning)
# ==========================================
K = 12
seeds = rng.uniform(0, boxsize, size=(K, 3))


# ==========================================
# 6) Objective: simple &quot;success&quot; on small-r bins
# ==========================================
def mse_on_bins(xi_model, xi_target, r_centers, rmax=60.0):
    &quot;&quot;&quot;
    Mean squared error over bins with r &lt;= rmax (simple success criterion).
    &quot;&quot;&quot;
    m = r_centers &lt;= rmax
    return float(np.mean((xi_model[m] - xi_target[m]) ** 2))


# ==========================================
# 7) Mini-grid search over (lambda, sigma)
# ==========================================
sigmas = [6.0, 8.0, 10.0, 12.0, 15.0]   # seed width (correlation length scale)
lams   = [0.5, 1.0, 2.0, 3.0, 4.0, 5.0] # dilation strength

best = None

# Baseline xi (should be ~0)
DD0 = pair_counts_weighted(pos0, pos0, bins, boxsize, autocorr=True, chunk=512)
DR0 = pair_counts_weighted(pos0, posR, bins, boxsize, autocorr=False, chunk=512)
xi_base = landy_szalay_xi(DD0, DR0, RR, nD=len(pos0), nR=len(posR))

for sigma in sigmas:
    phi = gaussian_seed_field(pos0, seeds, boxsize, sigma)

    for lam in lams:
        pos1 = epd_resample_catalog(pos0, phi, lam, rng)

        DD1 = pair_counts_weighted(pos1, pos1, bins, boxsize, autocorr=True, chunk=512)
        DR1 = pair_counts_weighted(pos1, posR, bins, boxsize, autocorr=False, chunk=512)
        xi_epd = landy_szalay_xi(DD1, DR1, RR, nD=len(pos1), nR=len(posR))

        score = mse_on_bins(xi_epd, xi_obs, r_centers, rmax=60.0)

        if best is None or score &lt; best[0]:
            best = (score, sigma, lam, xi_epd)

best_score, best_sigma, best_lam, xi_best = best

# ==========================================
# 8) Plot baseline vs best EPD vs observed
# ==========================================
plt.figure()
plt.plot(r_centers, xi_base, marker='o', label=&quot;Mock baseline (uniform Poisson)&quot;)
plt.plot(r_centers, xi_best, marker='o',
         label=f&quot;Best EPD-resampled (sigma={best_sigma}, lambda={best_lam})&quot;)
plt.plot(r_centers, xi_obs, marker='o', linestyle='--',
         label=&quot;Observed BOSS DR11 ξ₀(r) (reference)&quot;)
plt.axhline(0, linewidth=1)
plt.xlabel(&quot;r [Mpc/h]&quot;)
plt.ylabel(&quot;ξ(r) (monopole estimate)&quot;)
plt.title(&quot;EPD reweighting of mock catalogs: mini-grid tuning (toy demonstration)&quot;)
plt.legend()
plt.show()

print(&quot;=== Mini-grid search result ===&quot;)
print(f&quot;Best score (MSE on r&lt;=60): {best_score:.6e}&quot;)
print(f&quot;Best sigma: {best_sigma}  Best lambda: {best_lam}&quot;)
&lt;/syntaxhighlight&gt;

'''How to run:''' download the BOSS multipoles text file to your computer and set &lt;code&gt;path&lt;/code&gt; in the script to that local filename. Then run the script in Python (NumPy + Matplotlib required). The baseline curve should lie near &lt;math&gt;\xi\approx 0&lt;/math&gt;, while the EPD-resampled mock typically shows positive small-scale clustering.

= Scope and limitations =
EPD is a mathematical framework for measure transformations. It does not claim:
* a replacement theory for GR/QM,
* empirical confirmation without explicit predictions and tests.

== 8. Speculative Extensions and Geometric Renormalization ==
== Speculative Extensions and Geometric Renormalization ==

{{Disclaimer|This section is speculative and exploratory in nature.}}

Recent mathematical work extending Buffon's needle problem to curved manifolds suggests a possible connection between probability distributions and intrinsic geometry. In particular, studies of &quot;Buffon deficits&quot; on Gaussian manifolds indicate that deviations from the classical flat-space Buffon probability may encode Gaussian curvature through small geometric corrections.

In flat geometry, the classical Buffon crossing probability approaches:

&lt;math&gt;
\frac{2}{\pi}
&lt;/math&gt;

for sufficiently small needles.

On curved manifolds, however, the crossing probability acquires curvature-dependent corrections. This motivates a speculative extension of Einstein Probability Dilation (EPD): spacetime curvature may not only influence probability distributions, but probability structure itself may participate in the emergence or modulation of geometric behavior.

Within this interpretation, curvature may act as a statistical weighting mechanism on quantum path behavior. Rather than treating quantum path integrals strictly as sums over equally admissible histories, spacetime geometry could modulate the statistical contribution of classes of paths through curvature-dependent weighting functions.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x \, W(K)\, e^{iS/\hbar}
&lt;/math&gt;

where:

&lt;math&gt;e^{iS/\hbar}&lt;/math&gt;

represents the standard quantum probability amplitude,

&lt;math&gt;W(K)&lt;/math&gt;

represents a curvature-dependent geometric weighting factor,

and

&lt;math&gt;K&lt;/math&gt;

represents local Gaussian or spacetime curvature.

In this interpretation, the two probabilistic structures are not identical in mathematical character.

The Feynman factor

&lt;math&gt;e^{iS/\hbar}&lt;/math&gt;

is a complex quantum probability amplitude capable of interference and phase cancellation, whereas the curvature-weighting term behaves more like a real geometric measure acting on the statistical structure of path space itself.

Accordingly, EPD may not combine probabilities directly, but instead combine:
* a quantum interference structure,
* with a curvature-dependent probabilistic geometry.

The Buffon deficit framework suggests that curvature may appear as a probabilistic deviation from flat geometric behavior. Within EPD, this motivates the possibility that curvature-weighted probability measures could alter the statistical admissibility of quantum paths at extremely small scales.

If such curvature-weighting naturally suppresses pathological ultraviolet path contributions, the framework could potentially resemble a form of geometric renormalization in which divergence control emerges from geometric probability structure rather than purely algebraic subtraction procedures.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)\approx 1+cKL^2+O(L^3)
&lt;/math&gt;

where:

&lt;math&gt;L&lt;/math&gt;

represents a microscopic path scale,

&lt;math&gt;K&lt;/math&gt;

represents local curvature,

and

&lt;math&gt;c&lt;/math&gt;

represents a proportionality constant.

In the flat-space limit:

&lt;math&gt;
K\rightarrow0
&lt;/math&gt;

ordinary quantum behavior would be recovered.

At present these ideas remain heuristic and conceptual. However, they suggest a possible bridge between:
* stochastic geometry,
* probabilistic curvature,
* quantum path integrals,
* and emergent spacetime structure.

=== Open Questions ===

* Can curvature-weighted path measures be formally derived from probabilistic geometry?
* Can Buffon deficit relations be generalized from Gaussian manifolds to relativistic spacetime manifolds?
* Does curvature-weighting naturally regulate ultraviolet divergences in quantum field theory?
* What is the relationship between Gaussian curvature corrections and spacetime curvature tensors in General Relativity?
* Could probabilistic-geometric weighting contribute to emergent spacetime or stochastic gravity models?

= 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.

= See also =
* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Cosmological principle|Cosmological principle]]
* [[w:Holographic principle|Holographic principle]]
* [[w:Fractal geometry|Fractal geometry]]
* [[w:Quantum field theory|Quantum field theory]]
* [[w:Langlands program|Langlands program]]

= References =
==== Probability / measure theory ====
* [[w:Measure (mathematics)|Measure (mathematics)]]
* [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]]
* [[w:Probability theory|Probability theory]]&lt;ref&gt;Baryshnikov, Y., Cao, Y., Kahle, M., &amp; 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&lt;/ref&gt;
* &lt;ref&gt;Baryshnikov, Y., Cao, Y., Kahle, M., &amp; 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&lt;/ref&gt;

==== Computing / simulation ====
* [[w:Monte Carlo method|Monte Carlo method]]
* [[w:Importance sampling|Importance sampling]]

== Copyright and licensing ==
This Wikiversity page is authored by Howard Richardson. Please attribute appropriately when reusing or adapting this material.

Earlier and related versions may also be available on OSF and arXiv under the licenses stated on those platforms. Where multiple versions exist, the license shown on the specific platform/version governs reuse of that version.</text>
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    </revision>
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      <id>2810539</id>
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        <id>2995240</id>
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      <text bytes="25153" sha1="81riyx98qzv52l19a5k2bhgfvhrhsfj" xml:space="preserve">{{Research project}}
{{To be peer reviewed}}

= Einstein Probability Dilation (EPD) =

== Research abstract ==
'''Einstein Probability Dilation (EPD)''' 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.

EPD 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).

EPD 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 ==
EPD 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]]). EPD 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.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant: &lt;math&gt;Z(P,D)=\int_\Omega D\,dP&lt;/math&gt;.
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure, &lt;math&gt;\mathbb{E}_P[f]=\int_\Omega f\,dP&lt;/math&gt;.

== Probability dilation (reweighting) ==
Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D\,dP}{\int_\Omega D\,dP}\quad\text{for all }A\in\Sigma.&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt; where

&lt;math&gt;\widetilde{p}(x)=\frac{D(x)\,p(x)}{Z},\quad Z=\int_\Omega D(x)\,p(x)\,d\mu.&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

== Example: Lorentz contraction as a positive dilation field ==
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;\gamma(v)=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\quad\text{for } |v| &lt; c.&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;L(v)=\frac{L_0}{\gamma(v)}.&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;, for instance:
* &lt;math&gt;D(x)=\gamma(x)&lt;/math&gt; (weights states by increasing &lt;math&gt;\gamma&lt;/math&gt;), or
* &lt;math&gt;D(x)=\frac{1}{\gamma(x)}&lt;/math&gt; (weights states by decreasing &lt;math&gt;\gamma&lt;/math&gt;),

provided &lt;math&gt;D(x)&lt;/math&gt; is strictly positive over the model’s domain and &lt;math&gt;0&lt;\int_\Omega D\,dP&lt;\infty&lt;/math&gt;.

This example is included purely to illustrate “positive dilation”; it is not presented as a cosmological model.

=== Buffon sampling, the constant \pi, and a Lorentz-induced dilation factor (toy) ===
This brief example shows how &lt;math&gt;\pi&lt;/math&gt; naturally appears from sampling geometry (Buffon’s needle) and how inserting Lorentz contraction into the sampled length produces a strictly positive, dimensionless candidate dilation field. This is included for mathematical intuition; it is not claimed as a physical derivation.

For the classical [[w:Buffon's needle problem|Buffon needle problem]] with needle length &lt;math&gt;L\le d&lt;/math&gt; and line spacing &lt;math&gt;d&lt;/math&gt;, the crossing probability is:
&lt;math&gt;P_{\mathrm{cross}}(L;d)=\frac{2L}{\pi d}.&lt;/math&gt;

If a rest-length &lt;math&gt;L_0&lt;/math&gt; undergoes Lorentz contraction &lt;math&gt;L(v)=L_0/\gamma(v)&lt;/math&gt; for &lt;math&gt;|v| &lt; c&lt;/math&gt;, then the same form gives:
&lt;math&gt;P_{\mathrm{cross}}(v)=\frac{2L_0}{\pi d\,\gamma(v)}.&lt;/math&gt;

Normalizing to the rest case yields a dimensionless sampling ratio:
&lt;math&gt;D_{\pi}(v):=\frac{P_{\mathrm{cross}}(v)}{P_{\mathrm{cross}}(0)}=\frac{1}{\gamma(v)}=\sqrt{1-\frac{v^2}{c^2}}.&lt;/math&gt;

This &lt;math&gt;D_{\pi}(v)&lt;/math&gt; is strictly positive on the domain &lt;math&gt;|v| &lt; c&lt;/math&gt; and can be used (illustratively) as a dilation field in the EPD transform:
&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D_{\pi}(v)\,dP}{\int_\Omega D_{\pi}(v)\,dP}.&lt;/math&gt;

== Theorem-style mathematics expansion ==

=== Definition (EPD reweighting operator) ===
Let &lt;math&gt;P&lt;/math&gt; be a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; and &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; measurable with &lt;math&gt;0&lt;\int_\Omega D\,dP&lt;\infty&lt;/math&gt;. Define &lt;math&gt;\mathrm{EPD}(P;D)=\widetilde{P}&lt;/math&gt; by:

&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D\,dP}{\int_\Omega D\,dP}.&lt;/math&gt;

=== Theorem 1 (Normalization and positivity) ===
&lt;math&gt;\widetilde{P}&lt;/math&gt; is a probability measure. In particular:
# &lt;math&gt;\widetilde{P}(A)\ge 0&lt;/math&gt; for all measurable &lt;math&gt;A&lt;/math&gt;,
# &lt;math&gt;\widetilde{P}(\Omega)=1&lt;/math&gt;,
# If &lt;math&gt;P(A)=0&lt;/math&gt; then &lt;math&gt;\widetilde{P}(A)=0&lt;/math&gt; (absolute continuity preserved).

=== Theorem 2 (Expectation reweighting identity) ===
If &lt;math&gt;f&lt;/math&gt; is integrable under &lt;math&gt;\widetilde{P}&lt;/math&gt;, then

&lt;math&gt;\mathbb{E}_{\widetilde{P}}[f]=\frac{\mathbb{E}_P[D\,f]}{\mathbb{E}_P[D]}.&lt;/math&gt;

=== Theorem 3 (Composition / iteration rule) ===
Let &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D_1)&lt;/math&gt; and &lt;math&gt;\widehat{P}=\mathrm{EPD}(\widetilde{P};D_2)&lt;/math&gt;. Then

&lt;math&gt;\widehat{P}=\mathrm{EPD}(P;D_1D_2)&lt;/math&gt;

provided &lt;math&gt;\int_\Omega D_1D_2\,dP&lt;/math&gt; is finite and nonzero.

=== Theorem 4 (Fixed-point condition) ===
&lt;math&gt;P&lt;/math&gt; is a fixed point of EPD under dilation &lt;math&gt;D&lt;/math&gt; (i.e., &lt;math&gt;\mathrm{EPD}(P;D)=P&lt;/math&gt;) if and only if &lt;math&gt;D&lt;/math&gt; is constant &lt;math&gt;P&lt;/math&gt;-almost everywhere.

== Invariant quantities ==
EPD naturally highlights quantities that are invariant (or transform in controlled ways) under reweighting, including:
* ratios of expectations of the form &lt;math&gt;\mathbb{E}_P[D f]/\mathbb{E}_P[D]&lt;/math&gt;,
* equivalence classes of observables that remain unchanged under specific families of dilations,
* fixed-point structure (trivial dilations) and near-fixed behavior (small perturbations).

= Core axioms of Einstein Probability Dilation =

=== Axiom 1 — Probability primacy ===
Probability measures are treated as primary mathematical objects; structure is inferred from invariant properties of measure transformations.

=== Axiom 2 — Positive dilation ===
Dilation fields are positive (&lt;math&gt;D&lt;/math&gt; is strictly positive) so that reweighting preserves positivity and supports a consistent normalization constant.

=== Axiom 3 — Iterative composability ===
Successive dilations compose multiplicatively (&lt;math&gt;D_1D_2&lt;/math&gt;), enabling multi-step (iterative) dynamics on measures.

= Falsifiability conditions =
A proposed EPD-based model should specify:
* the configuration space &lt;math&gt;\Omega&lt;/math&gt;,
* a baseline measure &lt;math&gt;P&lt;/math&gt;,
* an explicit dilation field &lt;math&gt;D&lt;/math&gt; (with parameters and domain),
* measurable predictions (statistics, scaling laws, invariants) that can be checked against simulation or data,
* a baseline comparison (null model) and evaluation criteria (error measures, goodness-of-fit, or hypothesis tests).

'''Concrete falsifiability test (clustering example):''' Choose a baseline mock galaxy catalog (configuration space &lt;math&gt;\Omega&lt;/math&gt;), define an explicit strictly positive dilation field &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt;, apply EPD-style reweighting (e.g., importance resampling of the empirical measure), and compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt; using a standard estimator (e.g., normalized Landy–Szalay). Define a quantitative success metric (for example, mean-squared error between modeled and observed &lt;math&gt;\xi(r)&lt;/math&gt; over specified separation bins such as &lt;math&gt;r\le 60\,\mathrm{Mpc}/h&lt;/math&gt;). The model is falsified if no parameter choices within the stated domain can meet the pre-registered threshold under the chosen metric, or if the fit fails out-of-sample on withheld bins.

= Numerical simulation and iterative models =

== Simulation model description ==
In discrete demonstrations, the “state space” may be a finite set (e.g., bins, configurations, or catalog points), and the probability measure &lt;math&gt;P&lt;/math&gt; may be represented as a probability vector (or an empirical measure) over those states.

Two equivalent discrete implementations are common:
* '''weighted evaluation''': keep all points and carry weights proportional to &lt;math&gt;D&lt;/math&gt;, or
* '''importance resampling''': draw a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs and comparing \xi(r) ===
This demonstration connects the EPD reweighting idea to a standard observable in large-scale structure: the (monopole) two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;. The goal here is not a precision cosmology fit, but a clean, falsifiable pipeline:

# generate a baseline mock galaxy catalog in a periodic box (approximately unclustered),
# define a strictly positive dilation field &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt; over galaxy positions,
# perform EPD-style importance resampling to obtain a reweighted (clustered) mock catalog,
# measure &lt;math&gt;\xi(r)&lt;/math&gt; using the properly normalized Landy–Szalay estimator,
# tune a small mini-grid over &lt;math&gt;(\sigma,\lambda)&lt;/math&gt; to minimize a simple mismatch score on small scales.

==== Key definitions ====
* Dilation field: &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt;, with &lt;math&gt;\lambda &gt; 0&lt;/math&gt; and &lt;math&gt;\phi(x)\ge 0&lt;/math&gt;.
* Example potential: &lt;math&gt;\phi(x)=\sum_k \exp\!\left(-\|x-s_k\|^2/(2\sigma^2)\right)&lt;/math&gt; using seed points &lt;math&gt;s_k&lt;/math&gt;.
* Landy–Szalay estimator (normalized counts):
  &lt;math&gt;\xi(r)=\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}&lt;/math&gt;,
  where &lt;math&gt;DD,DR,RR&lt;/math&gt; are pair counts normalized by the total number of possible pairs in each catalog.

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
import numpy as np
import matplotlib.pyplot as plt

# ==========================================
# 1) Pair-counting with periodic boundaries
# ==========================================
def pair_counts_weighted(posA, posB, bins, boxsize, wA=None, wB=None,
                         autocorr=False, chunk=512):
    &quot;&quot;&quot;
    Weighted pair counts in radial bins with periodic boundary conditions.
    - autocorr=True: count unique pairs i&lt;j within posA
    - autocorr=False: count all cross pairs between posA and posB
    Returns raw (un-normalized) weighted counts per bin.
    &quot;&quot;&quot;
    posA = np.asarray(posA, float)
    posB = np.asarray(posB, float)
    bins = np.asarray(bins, float)
    nb = len(bins) - 1
    L = float(boxsize)

    if wA is None:
        wA = np.ones(len(posA), float)
    if wB is None:
        wB = np.ones(len(posB), float)
    wA = np.asarray(wA, float)
    wB = np.asarray(wB, float)

    counts = np.zeros(nb, float)

    if autocorr:
        N = len(posA)
        for i0 in range(0, N, chunk):
            i1 = min(N, i0 + chunk)
            A = posA[i0:i1]
            wAi = wA[i0:i1]

            # distances from A to ALL posA (periodic)
            d = A[:, None, :] - posA[None, :, :]
            d -= L * np.round(d / L)
            r = np.sqrt(np.sum(d * d, axis=2))

            W = wAi[:, None] * wA[None, :]

            # accumulate only i&lt;j
            for ii in range(i1 - i0):
                j_start = i0 + ii + 1
                if j_start &gt;= N:
                    continue
                rr = r[ii, j_start:]
                ww = W[ii, j_start:]
                hist, _ = np.histogram(rr, bins=bins, weights=ww)
                counts += hist
        return counts

    else:
        NA = len(posA)
        for i0 in range(0, NA, chunk):
            i1 = min(NA, i0 + chunk)
            A = posA[i0:i1]
            wAi = wA[i0:i1]

            d = A[:, None, :] - posB[None, :, :]
            d -= L * np.round(d / L)
            r = np.sqrt(np.sum(d * d, axis=2))

            W = wAi[:, None] * wB[None, :]
            hist, _ = np.histogram(r, bins=bins, weights=W)
            counts += hist
        return counts


def landy_szalay_xi(DD_raw, DR_raw, RR_raw, nD, nR):
    &quot;&quot;&quot;
    Properly normalized Landy–Szalay estimator:
      DD = DD_raw / [nD (nD-1)/2]
      DR = DR_raw / [nD nR]
      RR = RR_raw / [nR (nR-1)/2]
      xi = (DD - 2 DR + RR) / RR
    &quot;&quot;&quot;
    DD = DD_raw / (nD * (nD - 1) / 2.0)
    DR = DR_raw / (nD * nR)
    RR = RR_raw / (nR * (nR - 1) / 2.0)
    return (DD - 2 * DR + RR) / np.maximum(RR, 1e-30)


# ==========================================
# 2) EPD-style dilation field + resampling
# ==========================================
def gaussian_seed_field(pos, seeds, boxsize, sigma):
    &quot;&quot;&quot;
    phi(x) = sum_k exp(-||x-seed_k||^2/(2 sigma^2)) with periodic distances.
    Always nonnegative.
    &quot;&quot;&quot;
    pos = np.asarray(pos, float)
    seeds = np.asarray(seeds, float)
    L = float(boxsize)

    d = pos[:, None, :] - seeds[None, :, :]
    d -= L * np.round(d / L)
    r2 = np.sum(d * d, axis=2)
    phi = np.exp(-0.5 * r2 / (sigma ** 2)).sum(axis=1)
    return phi


def epd_resample_catalog(pos0, phi, lam, rng):
    &quot;&quot;&quot;
    Importance-resampling version of discrete EPD on an empirical measure:
      weights ∝ D(x)=exp(lam * phi(x))
      resample points with prob ∝ weights
    &quot;&quot;&quot;
    logw = lam * phi
    logw -= np.max(logw)          # stabilize
    w = np.exp(logw)              # strictly positive
    p = w / w.sum()
    idx = rng.choice(len(pos0), size=len(pos0), replace=True, p=p)
    return pos0[idx]


# ==========================================
# 3) Load real xi0(r) for reference (BOSS)
# ==========================================
# NOTE: Set this path to wherever you saved the file locally.
# The example below matches the earlier DR11 multipoles file name.
path = &quot;samushia_2013_CMASSDR11_xi_multipoles.dat&quot;
vals = np.loadtxt(path)
xi_obs = vals[:16]  # monopole xi0

# BOSS bins: edges 24..152 step 8 =&gt; centers 28..148
bins = np.arange(24, 152 + 8, 8)
r_centers = (bins[:-1] + bins[1:]) / 2


# ==========================================
# 4) Build baseline mock + randoms (periodic box)
# ==========================================
rng = np.random.default_rng(7)

boxsize = 200.0
N_data0 = 1400
N_rand = 4500

pos0 = rng.uniform(0, boxsize, size=(N_data0, 3))
posR = rng.uniform(0, boxsize, size=(N_rand, 3))

# Precompute RR once (random-random doesn't change during grid search)
RR = pair_counts_weighted(posR, posR, bins, boxsize, autocorr=True, chunk=512)


# ==========================================
# 5) Define seed locations (kept fixed during tuning)
# ==========================================
K = 12
seeds = rng.uniform(0, boxsize, size=(K, 3))


# ==========================================
# 6) Objective: simple &quot;success&quot; on small-r bins
# ==========================================
def mse_on_bins(xi_model, xi_target, r_centers, rmax=60.0):
    &quot;&quot;&quot;
    Mean squared error over bins with r &lt;= rmax (simple success criterion).
    &quot;&quot;&quot;
    m = r_centers &lt;= rmax
    return float(np.mean((xi_model[m] - xi_target[m]) ** 2))


# ==========================================
# 7) Mini-grid search over (lambda, sigma)
# ==========================================
sigmas = [6.0, 8.0, 10.0, 12.0, 15.0]   # seed width (correlation length scale)
lams   = [0.5, 1.0, 2.0, 3.0, 4.0, 5.0] # dilation strength

best = None

# Baseline xi (should be ~0)
DD0 = pair_counts_weighted(pos0, pos0, bins, boxsize, autocorr=True, chunk=512)
DR0 = pair_counts_weighted(pos0, posR, bins, boxsize, autocorr=False, chunk=512)
xi_base = landy_szalay_xi(DD0, DR0, RR, nD=len(pos0), nR=len(posR))

for sigma in sigmas:
    phi = gaussian_seed_field(pos0, seeds, boxsize, sigma)

    for lam in lams:
        pos1 = epd_resample_catalog(pos0, phi, lam, rng)

        DD1 = pair_counts_weighted(pos1, pos1, bins, boxsize, autocorr=True, chunk=512)
        DR1 = pair_counts_weighted(pos1, posR, bins, boxsize, autocorr=False, chunk=512)
        xi_epd = landy_szalay_xi(DD1, DR1, RR, nD=len(pos1), nR=len(posR))

        score = mse_on_bins(xi_epd, xi_obs, r_centers, rmax=60.0)

        if best is None or score &lt; best[0]:
            best = (score, sigma, lam, xi_epd)

best_score, best_sigma, best_lam, xi_best = best

# ==========================================
# 8) Plot baseline vs best EPD vs observed
# ==========================================
plt.figure()
plt.plot(r_centers, xi_base, marker='o', label=&quot;Mock baseline (uniform Poisson)&quot;)
plt.plot(r_centers, xi_best, marker='o',
         label=f&quot;Best EPD-resampled (sigma={best_sigma}, lambda={best_lam})&quot;)
plt.plot(r_centers, xi_obs, marker='o', linestyle='--',
         label=&quot;Observed BOSS DR11 ξ₀(r) (reference)&quot;)
plt.axhline(0, linewidth=1)
plt.xlabel(&quot;r [Mpc/h]&quot;)
plt.ylabel(&quot;ξ(r) (monopole estimate)&quot;)
plt.title(&quot;EPD reweighting of mock catalogs: mini-grid tuning (toy demonstration)&quot;)
plt.legend()
plt.show()

print(&quot;=== Mini-grid search result ===&quot;)
print(f&quot;Best score (MSE on r&lt;=60): {best_score:.6e}&quot;)
print(f&quot;Best sigma: {best_sigma}  Best lambda: {best_lam}&quot;)
&lt;/syntaxhighlight&gt;

'''How to run:''' download the BOSS multipoles text file to your computer and set &lt;code&gt;path&lt;/code&gt; in the script to that local filename. Then run the script in Python (NumPy + Matplotlib required). The baseline curve should lie near &lt;math&gt;\xi\approx 0&lt;/math&gt;, while the EPD-resampled mock typically shows positive small-scale clustering.

= Scope and limitations =
EPD is a mathematical framework for measure transformations. It does not claim:
* a replacement theory for GR/QM,
* empirical confirmation without explicit predictions and tests.


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 ==

Recent mathematical work extending Buffon's needle problem to curved manifolds suggests a possible connection between probability distributions and intrinsic geometry. In particular, studies of &quot;Buffon deficits&quot; on Gaussian manifolds indicate that deviations from the classical flat-space Buffon probability may encode Gaussian curvature through small geometric corrections.

In flat geometry, the classical Buffon crossing probability approaches:

&lt;math&gt;
\frac{2}{\pi}
&lt;/math&gt;

for sufficiently small needles.

On curved manifolds, however, the crossing probability acquires curvature-dependent corrections. This motivates a speculative extension of Einstein Probability Dilation (EPD): spacetime curvature may not only influence probability distributions, but probability structure itself may participate in the emergence or modulation of geometric behavior.

Within this interpretation, curvature may act as a statistical weighting mechanism on quantum path behavior. Rather than treating quantum path integrals strictly as sums over equally admissible histories, spacetime geometry could modulate the statistical contribution of classes of paths through curvature-dependent weighting functions.&lt;ref&gt;Feynman, R. P., &amp; Hibbs, A. R. (1965). ''Quantum Mechanics and Path Integrals''. McGraw-Hill.&lt;/ref&gt;

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x \, W(K)\, e^{iS/\hbar}
&lt;/math&gt;

where:

&lt;math&gt;e^{iS/\hbar}&lt;/math&gt;

represents the standard quantum probability amplitude,

&lt;math&gt;W(K)&lt;/math&gt;

represents a curvature-dependent geometric weighting factor,

and

&lt;math&gt;K&lt;/math&gt;

represents local Gaussian or spacetime curvature.

In this interpretation, the two probabilistic structures are not identical in mathematical character.

The Feynman factor

&lt;math&gt;e^{iS/\hbar}&lt;/math&gt;

is a complex quantum probability amplitude capable of interference and phase cancellation, whereas the curvature-weighting term behaves more like a real geometric measure acting on the statistical structure of path space itself.

Accordingly, EPD may not combine probabilities directly, but instead combine:
* a quantum interference structure,
* with a curvature-dependent probabilistic geometry.

The Buffon deficit framework suggests that curvature may appear as a probabilistic deviation from flat geometric behavior. Within EPD, this motivates the possibility that curvature-weighted probability measures could alter the statistical admissibility of quantum paths at extremely small scales.

If such curvature-weighting naturally suppresses pathological ultraviolet path contributions, the framework could potentially resemble a form of geometric renormalization in which divergence control emerges from geometric probability structure rather than purely algebraic subtraction procedures.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)\approx 1+cKL^2+O(L^3)
&lt;/math&gt;

where:

&lt;math&gt;L&lt;/math&gt;

represents a microscopic path scale,

&lt;math&gt;K&lt;/math&gt;

represents local curvature,

and

&lt;math&gt;c&lt;/math&gt;

represents a proportionality constant.

In the flat-space limit:

&lt;math&gt;
K\rightarrow0
&lt;/math&gt;

ordinary quantum behavior would be recovered.

At present these ideas remain heuristic and conceptual. However, they suggest a possible bridge between:
* stochastic geometry,
* probabilistic curvature,
* quantum path integrals,
* and emergent spacetime structure.
==== Open Questions ====

* Can curvature-weighted path measures be formally derived?
* Can Buffon deficit relations be generalized to relativistic spacetime?
* Could curvature-weighting regulate ultraviolet divergences?
* What relationship exists between Gaussian curvature and spacetime curvature tensors?
* Could probabilistic geometry contribute to emergent spacetime models?

= 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.

= See also =
* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Cosmological principle|Cosmological principle]]
* [[w:Holographic principle|Holographic principle]]
* [[w:Fractal geometry|Fractal geometry]]
* [[w:Quantum field theory|Quantum field theory]]
* [[w:Langlands program|Langlands program]]

== Status of the Framework ==
Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model. Several portions of the article discuss heuristic interpretations and proposed extensions which are not established physical theory.

The article is intended as an educational and exploratory investigation into possible relationships between probability structure, geometric curvature, and quantum path behavior.

= References =
==== Probability / measure theory ====
* [[w:Measure (mathematics)|Measure (mathematics)]]
* [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]]
* [[w:Probability theory|Probability theory]]&lt;ref&gt;Baryshnikov, Y., Cao, Y., Kahle, M., &amp; 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&lt;/ref&gt;
* &lt;ref&gt;Baryshnikov, Y., Cao, Y., Kahle, M., &amp; 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&lt;/ref&gt;

==== Computing / simulation ====
* [[w:Monte Carlo method|Monte Carlo method]]
* [[w:Importance sampling|Importance sampling]]

== Copyright and licensing ==
This Wikiversity page is authored by Howard Richardson. Please attribute appropriately when reusing or adapting this material.

Earlier and related versions may also be available on OSF and arXiv under the licenses stated on those platforms. Where multiple versions exist, the license shown on the specific platform/version governs reuse of that version.</text>
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    <title>Republic of Ireland Legal System</title>
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      <text bytes="88925" sha1="kk638fuvr54yclf29plhrnf47cn5d0d" xml:space="preserve">== '''Introduction and Sources of Communication Law''' ==

=== Irish History and the Role of Catholicism ===
English settlers began coming to Ireland around 1169 and brought the common law system with them.&lt;ref&gt;Janet Sinder, ''Irish Legal History: An Overview and Guide to the Sources'', 93 Law Library Journal 231, 243 (2001).&lt;/ref&gt;  At the time, the Irish were governed by the brehon law system, which consisted of laws that were “the written version of oral rules that had been passed down for generations,” rather than “statutes enacted by a legislative body.”&lt;ref&gt;''Id''. at 238.&lt;/ref&gt;  The common law system brought by the English and the brehon system existed together in the State for almost 500 years, until brehon law was suppressed with the Tudor conquest and later outlawed.&lt;ref&gt;''Id''. at 243, 248.&lt;/ref&gt;  The English took control of Ireland and developed the legal system, enacting statutes to address social problems that existed because of unrest in Ireland following the English conquest.&lt;ref&gt;''Id''. at 249.&lt;/ref&gt;  Specifically, “penal laws” were passed with the intent to “convert the Irish to Protestantism and to penalize them if they remained Catholic.”&lt;ref&gt;''Id''. at 249.&lt;/ref&gt;  

In 1800, the Irish Parliament agreed to the Act of the Union, making Ireland a part of the United Kingdom, adopting practically the same legal system as that of Great Britain.&lt;ref&gt;''Id''. at 250.&lt;/ref&gt;  In 1919, the Anglo-Irish War began and Ireland established its own local government known as “The Dáil” and in 1920, the Government of Ireland Act divided Ireland into two regions – Southern and Northern Ireland.&lt;ref&gt;''Id''. at 251-52.&lt;/ref&gt;  Following a truce, the war ended and the Constitution of the Irish Free State was drafted, a departure from the British legal system which has no written constitution.&lt;ref&gt;''Id''. at 252-53.&lt;/ref&gt;  The South remained largely Catholic, while Northern Ireland was primarily Protestant.&lt;ref&gt;Gerard Whyte, ''Religion and the Irish Constitution'', 30 J. Marshall L. Rev. 725, 727 (1997).&lt;/ref&gt;  Following the British conquest of the State, a new Constitution was adopted in 1937 which reflected the current ideologies in Ireland.&lt;ref&gt;''Id''. at 728.&lt;/ref&gt;  Ireland’s history is important in understanding the underlying themes of Ireland’s current communication law, as two notable influences of the Constitution come from “the inherited liberal democratic tradition of the Free State” and “the social teaching of the Catholic Church.”&lt;ref&gt;''Id''.&lt;/ref&gt;

=== Structure of the Irish Legal System ===
Ireland is a parliamentary republic with power divided among three branches of government, as laid out by the Constitution.&lt;ref&gt;Northwestern Pritzker School of Law, “Ireland: International Team Project, Government Structure,” [https://library.law.northwestern.edu/c.php?g=1369634&amp;p=10120307#:~:text=News%20and%20Statistics-,Government%20Structure,interprets%20and%20applies%20the%20laws. https://library.law.northwestern.edu/c.php?g=1369634&amp;p=10120307#:~:text=News%20and%20Statistics-,Government%20Structure,interprets%20and%20applies%20the%20laws.]&lt;/ref&gt;  The legislative power is vested in the Oireachtas, which consists of the President and the two houses of the Oireachtas, the Dáil and the Seanad.&lt;ref&gt;Houses of the Oireachtas, “Factsheet,” https://www.oireachtas.ie/en/press-centre/factsheet/.&lt;/ref&gt;  The executive power is given to the Government, which is headed by the Prime Minister.&lt;ref&gt;''Id''.&lt;/ref&gt;  The judicial power is vested in the court system, which is made up of five main courts.&lt;ref&gt;Northwestern Pritzker School of Law, ''supra'' n. 12; Citizens Information, “Courts,” https://www.citizensinformation.ie/en/justice/courts-system/courts/.&lt;/ref&gt;

=== '''Sources of Communication Law in Ireland''' ===
Ireland’s legal system operates through three main sources of law: The Constitution, the Common Law, and Statutes.&lt;ref&gt;Stephen Walsh &amp; Co. Solicitors, “The Irish Legal System Explained,” https://stephenwalshsolicitors.ie/irish-legal-system-explained/#:~:text=Ireland's%20legal%20system%20has%20three,law%20of%20the%20European%20Union.&lt;/ref&gt;  Ireland joined the European Communities, now the European Union (“EU”) in 1973, and is also subscribed to EU law.&lt;ref&gt;''Id''.; European Union, Representation in Ireland, “Ireland in the EU,” https://ireland.representation.ec.europa.eu/about-us/irelands-eu-membership_en.&lt;/ref&gt;. Communication law in Ireland is shaped by a combination of domestic, regional, and international sources of law.  The primary foundations lie in the Constitution, which protects fundamental freedoms.  Legislation enacted by the Oireachtas includes statutes regulating broadcasting, media, and digital communications.  In addition, European Union law and the ECHR play a significant role in ensuring Irish legislation complies with relevant regulations and guidelines.

==== The Constitution ====
The [https://www.irishstatutebook.ie/pdf/ga.cons.pdf Irish Constitution] was ratified in 1937 and “is the fundamental law of the state.”&lt;ref&gt;Stephen Walsh &amp; Co. Solicitors, ''supra'' n. 16.&lt;/ref&gt;  The Constitution lays out the separation of powers and functions of the State, and provides the fundamental rights guaranteed to Irish citizens.&lt;ref&gt;''Id''.&lt;/ref&gt;  Specifically, the Constitution guarantees citizens the right to “express freely their convictions and opinions.”&lt;ref&gt;Constitution of Ireland, Article 40.6.1(i).&lt;/ref&gt;  This grant of free expression, however, contains a limit on expression that “undermines public order or morality or the authority of the State” and specifically states that “[t]he publication or utterance of seditious or indecent matter is an offence . . . punishable [by] law.”&lt;ref&gt;''Id''.&lt;/ref&gt;

Prior to the adoption of the 1937 Irish Constitution, common law principles from England were embedded in Irish law.&lt;ref&gt;David Kenny, ''God in the Irish Constitution,'' BYU Law (Oct. 31, 2020), https://talkabout.iclrs.org/2020/10/31/god-in-the-irish-constitution/.&lt;/ref&gt;  When the 1937 Constitution was adopted, it was influenced heavily by both common law principles that previously existed, as well as and Catholic principles and traditions.&lt;ref&gt;''Id''.&lt;/ref&gt; Specifically, the Constitution was, in part, an attempt “to unite republican principle with a confessional infusion of Roman Catholic thinking and teaching on the nature of the state, the common good, and the place of the rights of the individual.”&lt;ref&gt;Kevin Boyle, “The Irish Constitution and the Rights of the Individual,” Article 19 Research and Information Centre on Censorship, London, [https://www.persee.fr/doc/irlan_0183-973x_1988_num_13_2_2821. https://www.persee.fr/doc/irlan_0183-973x_1988_num_13_2_2821.]&lt;/ref&gt;  The extent to which Catholicism shaped the Constitution can be seen through the inclusion of religious elements in much of the document.&lt;ref&gt;''Id''.&lt;/ref&gt;

==== Domestic Legislation ====
The [https://www.cnam.ie/app/uploads/2024/11/Coimisiun-na-Mean-Online-Safety-Code.pdf Online Safety Code] was developed by the Coimisiún na Meán and is a mandatory set of rules that must be followed by video-sharing platforms.&lt;ref&gt;Coimisiún na Meán, “Legislation Explained,” [https://www.cnam.ie/general-public/online-safety/online-safety-framework/legislation-explained/. https://www.cnam.ie/general-public/online-safety/online-safety-framework/legislation-explained/.]&lt;/ref&gt;  The rules prohibit illegal context, including “child sexual abuse materials, child trafficking, terrorist content, offences concerning racism/xenophobia or incitement of violence or hatred against a group of persons.”&lt;ref&gt;''Id''.&lt;/ref&gt;  The Code also prohibits harmful context, such as “cyberbullying, pornography, gross violence, promoting eating disorders, and promoting suicide or self-harm content.”&lt;ref&gt;''Id''.&lt;/ref&gt;

The [https://revisedacts.lawreform.ie/eli/2009/act/18/revised/en/pdf?annotations=true Broadcasting Act 2009], as amended by the Online Safety and Media Regulation Act 2022,  “determines the regulatory framework for the broadcasting, audiovisual on-demand, and online safety in Ireland,” and governs “issues relating to public service broadcasters, media development, and public funding.”&lt;ref&gt;Coimisiún na Meán, “Broadcasting Act 2009,” [https://www.cnam.ie/industry-and-professionals/codes-legislation/legislation/broadcasting-act/. https://www.cnam.ie/industry-and-professionals/codes-legislation/legislation/broadcasting-act/.]&lt;/ref&gt;

The Irish government has also announced approval of the General Scheme of the Media Regulation Bill, which will “give effect to the main elements” of the [https://www.irishstatutebook.ie/eli/2025/si/22/made/en/pdf European Media Freedom Act] (“EMFA”) in Ireland.&lt;ref&gt;Government of Ireland, Department of Culture, Communications and Sport, “Minister O’Donovan Secures Government Approval of the General Scheme of the Media Regulation Bill,” (Jul. 2 2025), https://www.gov.ie/en/department-of-culture-communications-and-sport/press-releases/minister-odonovan-secures-government-approval-of-the-general-scheme-of-the-media-regulation-bill/.&lt;/ref&gt;  The EMFA protects media independence via safeguards for journalist sources, protection against “unjustified online content removal,” and transparency for the media audience.&lt;ref&gt;Coimisiún na Meán, “European Media Freedom Act,” https://www.cnam.ie/industry-and-professionals/codes-legislation/legislation/european-media-freedom-act/.&lt;/ref&gt;

==== Relevant Regulatory Bodies ====
The Communications Regulation Act 2002 created the Commission for Communications Regulation (“COMREG”).&lt;ref&gt;Paul McMahon, “ComReg,” McMahon Legal Solicitors and Legal Consultants, https://mcmahonsolicitors.ie/comreg-2/#:~:text=The%20ODTR%20later%20became%20the,its%20powers%20under%20European%20regulations.&lt;/ref&gt;  The statutory body oversees “the regulation of the electronic communications sector (telecommunications, radio-communications and broadcasting transmission) and the postal sector.&lt;ref&gt;Commission for Communications Regulation, “Who We Are and What We Do,” https://www.comreg.ie/about/foi-aie-info/who-we-are-and-what-we-do/.&lt;/ref&gt;  COMREG has responsibilities in the telecommunications sector related to electronic communications services and networks, and posts and spectrum management.&lt;ref&gt;Commission for Communications Regulation, “What We Do,” https://www.comreg.ie/about/what-we-do/.&lt;/ref&gt;  The body is &quot;responsible for facilitating competition, for protecting consumers and for encouraging innovation.&quot;&lt;ref&gt;''Id''.&lt;/ref&gt;  COMREG operates under both Irish and EU legislation.&lt;ref&gt;''Id''.&lt;/ref&gt;  COMREG also &quot;has surveillance and enforcement responsibilities in relation to radio equipment regulation, and regulation of devices relating to electromagnetic compatibility.&quot;&lt;ref&gt;Commission for Communications Regulation, &quot;Who We Are and What We Do,&quot; https://www.comreg.ie/about/foi-aie-info/who-we-are-and-what-we-do/&lt;/ref&gt;. 

The Coimisiún na Meán is the “regulator of traditional and online media in Ireland,” as established by the Online Safety Media Regulation Act (“OSMR”).&lt;ref&gt;Coimisiún na Meán, “Legislation Explained,” https://www.cnam.ie/general-public/online-safety/online-safety-framework/legislation-explained/.&lt;/ref&gt;  The Coimisiún na Meán published an Online Safety Code and Online Safety Guidance Materials under the OSMR.&lt;ref&gt;Government of Ireland, Department of Culture, Communications and Sport, “Online Safety and Media Regulation Act 2022,” (last updated Apr. 15, 2025), https://www.gov.ie/en/department-of-culture-communications-and-sport/publications/online-safety-and-media-regulation-act-2022/.&lt;/ref&gt;. Prior to the establishment of the Coimisiún na Meán (&quot;Media Commission&quot;), the Broadcasting Authority of Ireland (&quot;BAI&quot;) was a primary media regulator in Ireland.&lt;ref&gt;John Cian McGrath and Kirsty Park, Dublin City University, “EDMO Policy Monitoring: The Regulation of Online Disinformation,” (Nov., 2022), [https://edmohub.ie/wp-content/uploads/2023/02/EDMO-Task-V-Dec-2022-2.pdf#:~:text=The%20Electoral%20Reform%20Act%20has%20been%20passed,not%20compatible%20with%20the%20e%2D%20Commerce%20Directive4. https://edmohub.ie/wp-content/uploads/2023/02/EDMO-Task-V-Dec-2022-2.pdf#:~:text=The%20Electoral%20Reform%20Act%20has%20been%20passed,not%20compatible%20with%20the%20e%2D%20Commerce%20Directive4.]&lt;/ref&gt;  The BAI was established by the Broadcasting Act of 2009.&lt;ref&gt;Communications Officer, &quot;Broadcasting Authority of Ireland (BAI),&quot; The Irish Film and Television Network, https://www.iftn.ie/rep_bodies/Agenices/?act1=record&amp;aid=16&amp;rid=64&amp;only=1.&lt;/ref&gt;.  The role of the BAI was &quot;to ensure that the number and categories of broadcasting services made available in the State best serve the needs of the people of the island of Ireland; to uphold the democratic values enshrined in the Constitution especially those relating to rightful liberty of expression and; to provide for open and pluralistic broadcasting services.&quot;&lt;ref&gt;''Id''.&lt;/ref&gt;  Those objectives were absorbed through the three-year transition integrating the BAI into the new Media Commission.&lt;ref&gt;McGrath and Park, ''supra'' note 32.&lt;/ref&gt;  BAI was fully dissolved in in March 2023 and the Media Commission became fully operational.&lt;ref&gt;Kate McKenna and Simon Shinkwin, “In brief: Telecoms Regulation in Ireland,” Matheson, (June 11, 2025), https://www.lexology.com/library/detail.aspx?g=62433d50-c5b1-4cb0-b3f9-5823363d35b5.&lt;/ref&gt;  The Media Commission is also deemed the Digital Services Coordinator for Ireland.&lt;ref&gt;''Id''.&lt;/ref&gt;

The Competition and Consumer Protection Commission (“CCPC”) is a regulatory authority which works with the Media Commission to enforce the Digital Services Act.&lt;ref&gt;''Id''.&lt;/ref&gt;

==== Regional Law ====
Ireland signed the European Convention on Human Rights (&quot;ECHR&quot;) in 1950 and then ratified it in 1953.&lt;ref&gt;Government of Ireland, Department of Foreign Affairs and Trade, “European Court of Human Rights,” (last updated Aug. 6, 2025), https://www.gov.ie/en/department-of-foreign-affairs/publications/european-court-of-human-rights/#:~:text=Ireland%20signed%20the%20Convention%20on,compulsory%20jurisdiction%20of%20the%20Court.&lt;/ref&gt; &lt;sup&gt; &lt;/sup&gt;Ireland later incorporated the Convention into Irish law through the European Convention on Human Rights Act 2003.&lt;ref&gt;''Id''.&lt;/ref&gt; &lt;sup&gt; &lt;/sup&gt; Specifically, Article 10 of the ECHR “gives you the right to freely express your opinions and views,” but “the right can be restricted or limited as long as this is allowed by law for a legitimate purpose and is done in a proportionate way.”&lt;ref&gt;Irish Human Rights and Equality Commission, “Human Rights and Constitutional Rights,” https://www.ihrec.ie/factsheets/human-rights-and-constitutional-rights.&lt;/ref&gt;  Although the European Court of Human Rights (“ECtHR”) can hear complaints of alleged violations of the ECHR, the case must first be brought within the Irish legal system and appealed as high as possible before it can be brought to the European Court of Human Rights.&lt;ref&gt;''Id''. The ECtHR is an international court which “consists of a judge from each EU member state” and “considers allegations that states have breached or failed to meet the requirements of the ECHR.” ''Id''.&lt;/ref&gt;

As an EU member, Ireland also enforces the [https://www.eu-digital-services-act.com/Digital_Services_Act_Articles.html Digital Services Act] (“DSA”), which is a set of rules that “applies to all online intermediary service providers (“ISPs”) that provide services in the EU.”&lt;ref&gt;Coimisiún na Meán, “Legislation Explained,” ''supra'' n. 22.&lt;/ref&gt;

== '''Principles of Communication Law and the Media''' ==
The principles of communication law in Ireland are primarily grounded in the balancing of freedom of expression and the need to protect the interest of the public.  These principles can be seen in the governing regulatory framework, which seeks to regulate the telecommunications in Ireland across all platforms.  The Coimisiún na Meán is the primarily body responsible for regulating with broadcasting, online media, and audio visual services. ComReg is the body responsible for the communications sphere in Ireland.  Together with EU law, these bodies work to oversee the telecoms sector and ensure that providers follow governing telecommunication regulations.

=== Telecommunication Law: Regulatory Framework ===
The relevant governmental department which oversees the telecoms sector in Ireland is the Department of the Environment, Climate and Communications (DECC).&lt;ref&gt;Kate McKenna and Simon Shinkwin, “In brief: Telecoms Regulation in Ireland,” Matheson, (June 11, 2025), https://www.lexology.com/library/detail.aspx?g=62433d50-c5b1-4cb0-b3f9-5823363d35b5.&lt;/ref&gt;  The governmental department which oversees the media sector is the Department of Tourism, Culture, Arts, Gaeltacht, Sport and Media.&lt;ref&gt;''Id''.&lt;/ref&gt;  The regulator of telecommunications in the nation is ComReg. &lt;ref&gt;''Id''.&lt;/ref&gt; As a member of the EU, Ireland adopted the European Electronic Communications Code (“ECC”), which includes a set of rules “regulating electronic communications (telecoms) networks, telecoms services, and associated facilities and services.”&lt;ref&gt;EUR-Lex, “Electronic Communications Code,” Publications Office of the European Union, https://eur-lex.europa.eu/EN/legal-content/glossary/electronic-communications-code.html.&lt;/ref&gt;

ComReg manages the communications sphere in Ireland by requiring authorization and enforcing the relevant regulations.&lt;ref&gt;Citizens Information, “Regulation of Postal Services and Electronic Communications,” https://www.citizensinformation.ie/en/consumer/phone-internet-tv-and-postal-services/regulation-of-postal-services-phone-internet-radion-and-tv/.&lt;/ref&gt;  ComReg regulates the sectors by ensuring the communication providers have the necessary authorization, licensing, and registration necessary to provide their services to the country.&lt;ref&gt;''Id''.&lt;/ref&gt;  As regulator of the communications sector, ComReg must “make sure providers follow the rules through enforcement[;] investigate complaints received from the wider industry and consumers[;] prevent fraud and misuse of Irish numbers to protect both end-users and operators[; and] make sure network incidents (breach of security or loss of integrity that has a significant impact on the operation of networks and services) are properly managed by network operators.”&lt;ref&gt;''Id''.&lt;/ref&gt;

=== Telecommunication Law: Licensing Regime ===
The supply of communications services in Ireland is governed by the regime established in the European Union Electronic Communications Code (“ECC”).&lt;ref&gt;''See'' McKenna and Shinkwin, ''supra'' note 37.&lt;/ref&gt;  The ECC supplies “the general right to provide an electronic communications network (“ECN”) or an electronic communications service (“ECS”) (or both) provided certain conditions are complied with.”&lt;ref&gt;''Id''.&lt;/ref&gt;  ComReg is responsible for implementing the procedure for operators to obtain general authorizations to use an ENC or ECS.&lt;ref&gt;''Id''.&lt;/ref&gt;  The ECC regulations also dictate the conferring of rights to use numbers and radio spectrum.&lt;ref&gt;''Id''.&lt;/ref&gt;

==== Spectrum Licenses ====
The ECC also lays out the legal framework that governs ComReg’s administration of the radio frequency spectrum in Ireland. &lt;ref&gt;''Id''.&lt;/ref&gt; ComReg issues licenses on a neutral basis, but may restrict the “types of radio network of wireless access technology used for ECS where it is necessary” for non-discriminatory reasons, such as avoiding harmful interference or safeguarding the effective use of the spectrum.&lt;ref&gt;''Id''.&lt;/ref&gt;  ComReg published The Wireless Telegraphy Regulations 2014, which contains “guidelines for spectrum trading in the Radio Spectrum Policy Programme (“RSPP”) bands.&quot;&lt;ref&gt;''Id''.&lt;/ref&gt;

== '''Censorship and Violent Content''' ==
Censorship law in Ireland reflects this balanced approach between the modern constitutional freedoms and the Irish-Catholic values.  Censorship law was previously more restrictive in Ireland, but has become more relaxed as the country has moved away from such traditional rules. Hate speech is regulated in conformity with EU law, but there is tension regarding the balance between free speech and hate speech.

“Censorship laws set out rules about standards” for forms of communication which may be restricted or banned if they are “unacceptable, offensive, obscene, [or] likely to incited hatred or violence.”&lt;ref&gt;Citizens Information, “Censorship,” https://www.citizensinformation.ie/en/government-in-ireland/irish-constitution-1/censorship/.&lt;/ref&gt;  The strong influence of Catholicism in shaping the Irish nation during its fight for independence from the British resulted in ideas of public morality that were characterized by “conservative Irish-Catholic values.”&lt;ref&gt;Gracia Larsen-Schmidt, “The Moral Ideas of the Community: Censorship and Irish-Catholic Nation Building,” Nursing Clio, (Feb. 5 2025), https://nursingclio.org/2025/02/05/the-moral-ideas-of-the-community-censorship-and-irish-catholic-nation-building/.&lt;/ref&gt;  As a result, administrative agencies were established to enact censorship laws which “prevented Irish people from accessing materials that depicted, humanized, and communicated specific elements of human life relating to sexuality, desire and bodies . . . in the name of defending Catholic morals and national identity.”&lt;ref&gt;''Id''.&lt;/ref&gt;. 

=== Censorship of Publications ===
Ireland created the Censorship of Publications Board (“the Board”) through the [https://www.irishstatutebook.ie/eli/1929/act/21/enacted/en/print.html Censorship of Publications Act of 1929].&lt;ref&gt;''Id''. The Board consists of five voluntary members “who are appointed by the Minister for Justice.” ''Id''.&lt;/ref&gt;  Books and periodicals are brought to the Board by the members of the public or customs or excise officers for examination.&lt;ref&gt;''Id''.&lt;/ref&gt;  The Board can choose whether or not to ban the publication, prohibiting the sale, purchase, or distribution of the publication in Ireland.&lt;ref&gt;''Id''. Bans can be appealed to the Censorship of Publications Appeal Board, which can choose to uphold, cancel, or modify the prohibition. ''Id''.&lt;/ref&gt;  

The Censorship of Publications Act was viewed as being influenced by the Catholic Church and existing sentiments to protect Ireland from foreign influences.&lt;ref&gt;EBSCO, “History of Censorship in Ireland,” https://www.ebsco.com/research-starters/politics-and-government/history-censorship-ireland.&lt;/ref&gt; The Board was “aided by Catholic societies and customs officials” and “banned literature based on the recommendations of Catholic activists,” until the only literature available to the Irish reading public primarily focused on religion and Irish culture.&lt;ref&gt;''Id''.&lt;/ref&gt;  Although efforts have been made to abolish the Censorship Board, it actively banned books in Ireland as recently as 2015 and the Censorship of Publications Act remains law.&lt;ref&gt;''Id''. “The Fianna Fáil attempted to abolish the Censorship Board in 2013 but was not successful.” ''Id''.&lt;/ref&gt;

=== Censorship of Film ===
The Irish Film Classification Office (“IFCO”) is a statutory body established by the [https://www.irishstatutebook.ie/eli/1923/act/23/enacted/en/print.html Censorship of Films Act 1923].&lt;ref&gt;''Id''.&lt;/ref&gt;  The IFCO examines and certifies all films, videos, and DVDs intended for distribution in Ireland with the purpose of “provid[ing] the public and parents . . . with a modern and dependable system of classification that: protects children and young persons[;] has regard for freedom of expression[; and] has respect for the values of Irish society.”&lt;ref&gt;Irish Film Classification Office, “What We Do,” https://www.ifco.ie/en/ifco/pages/what_we_do.&lt;/ref&gt;  IFCO certifies films and DVDs for viewing based on various criteria and “provides guidelines on an appropriate age group for film,” but can choose to ban a film or DVD that it deems unfit for viewing.&lt;ref&gt;Citizens Information, “Censorship” at Censorship of Films, DVDs and Video Games.&lt;/ref&gt;

=== Censorship of the Internet ===
Ireland currently does not have any law which specifically covers internet censorship, but there is legislation which affects certain online content.&lt;ref&gt;''Id''. at Censorship of the Internet. The Child Trafficking and Pornography Act 1988 mandates that ISPs remove any content that is illegal under the Act. ''Id''. The European Union Copyright and Related Rights Regulations are used to prevent the illegal downloading of copyrighted material. ''Id''.&lt;/ref&gt;  In December 2022, the [https://www.irishstatutebook.ie/eli/2022/act/41/enacted/en/print.html Online Safety and Media Regulation Act], which created a media regulator for online safety and an Online Safety Commissioner that regulates online services, went into effect.&lt;ref&gt;''Id''.&lt;/ref&gt;  Additionally, the European Union has regulations in place that target harmful or illegal online content in which Ireland participates.&lt;ref&gt;''Id''. Ireland is a member of the EU Internet forum set up to prevent international terrorists from misusing the internet and removing illegal content. ''Id''.&lt;/ref&gt;

=== '''Hate Speech''' ===
The [https://www.irishstatutebook.ie/eli/1989/act/19/enacted/en/print.html Prohibition of Incitement to Hatred Act 1989] (“1989 Act”) prohibits “incitement to hatred on account of race, religion, nationality or sexual orientation.”&lt;ref&gt;Prohibition of Incitement to Hatred Act, 1989, https://www.irishstatutebook.ie/eli/1989/act/19/enacted/en/print.html.&lt;/ref&gt;  The Act outlaws multiple forms of communication, precluding “the use of words, behavior or the publication or distribution of material which is threatening, abusive or insulting and intended, or likely, to stir up hatred.”&lt;ref&gt;OSCE Office for Democratic Institutions and Human Rights, “Hate Crime Legislation in Ireland,” https://hatecrime.osce.org/hate-crime-legislation-ireland.&lt;/ref&gt;  The 1989 Act has been criticized as outdated and ineffective, especially in combatting hate speech that occurs online.&lt;ref&gt;Roisín McFadden, “Online Hate Speech in Ireland and the United Kingdom,” UCD SLS Journal 2022, https://issuu.com/ucdstudentlegalservicesjournal/docs/sls_full_version_with_covers/s/15371427.&lt;/ref&gt;  The prosecution of Patrick Kissane is the only significant case where a party was tried under the 1989 Act.&lt;ref&gt;John O’Mahony, “Man Cleared of Online Hatred Against Travellers,” Irish Examiner (Oct. 1 2011), https://www.irishexaminer.com/news/arid-20169325.html.&lt;/ref&gt;  Following an interaction with a traveller, Kissane set up a Facebook Group named “Promote the Use of Knacker Babies as Shark Bait,” with phrases implying that Traveller babies should be used as bait and for testing new drugs for viruses.&lt;ref&gt;''Id''. “Irish Travellers are a small indigenous ethnic minority” living between Ireland, the United Kingdom, and the United States that are commonly “subjected to discrimination and exclusion.” Martin Collins, “Anti-Traveller Racism Pervasive and Deep-Rooted,” INAR, https://inar.ie/anti-traveller-racism-pervasive-and-deep-rooted/.&lt;/ref&gt;  Although the judge found the comments to be “obnoxious and revolting,” he considered them one-off comments and dismissed the case.&lt;ref&gt;John O’Mahony, “Man Cleared of Online Hatred Against Travellers,” ''supra'' n. 55.&lt;/ref&gt;  This case is a reflection of how difficult prosecution under this law was given its strict standards.&lt;ref&gt;Roisín McFadden, “Online Hate Speech in Ireland and the United Kingdom,” ''supra'' n. 54.&lt;/ref&gt;  This is in contrast to relevant law in the UK, which prosecutes hate speech more severely.&lt;ref&gt;Roger Berkeley, Law &amp; Liberty, “The Speech Battle on the Emerald Isle,” (Jul. 3 2025), https://lawliberty.org/the-speech-battle-on-the-emerald-isle/.&lt;/ref&gt;

As an EU member, the Digital Services Act required the Irish Government to enact a Hate Speech Bill.&lt;ref&gt;''Id''.&lt;/ref&gt;  In 2022, the Hate Speech Bill was introduced and passed in the Dáil with, but was held up in the Seanad for two years, with many free speech activists and groups campaigning against the Bill over its strict provisions prohibiting “possessing the wrong sort of meme” and giving “police the right to confiscate devices and compel passwords on the mere suspicion of such possession.”&lt;ref&gt;''Id''.&lt;/ref&gt;  Compromises were made and on December 31, 2024, new hate crime legislation went into effect that only focused on increasing sentences.&lt;ref&gt;''Id''.&lt;/ref&gt;  The [https://www.irishstatutebook.ie/eli/2024/act/41/enacted/en/print.html Criminal Justice (“Hate Offences”) Act 2024] “provides for increased prison sentences for certain crimes, where proven to be motivated by hatred, or where hatred is demonstrated.&lt;ref&gt;''Id''.&lt;/ref&gt;  In addition to physical offences, such as assault of property damage, the Hate Offences Act also covers speech-related offences, including “threats to kill or cause serious harm aggravated by hatred” and “distribution or display in public place of material which is threatening, abusive, insulting or obscene aggravated by hatred.”&lt;ref&gt;''Id''.&lt;/ref&gt;  However, the EU has mandated that Ireland revive its hate speech bill in compliance with the Digital Services Act, but Ireland has responded that the Hate Offences Act is sufficient.&lt;ref&gt;''Id''.&lt;/ref&gt;  This reflects the tension between the very rigid censorship laws that were prevalent in Ireland and free speech.&lt;ref&gt;''Id''.&lt;/ref&gt;

== '''Truth, Honor, and Tolerance''' ==
Defamation law in Ireland reflects the balance between the constitutional guarantee of free expression with the protection of one's reputation.  Defamation law is primarily governed by statute and case law and provides remedies for damage to one's good name, as well as defenses such as truth and honest opinion.  Disinformation law, by contrast, is not governed by one sole regulatory structure and involves the use of multiple regulations which address different areas of communication.  Together, the law's show Ireland's commitment to balancing constitutional freedoms of free speech and expression with public and individual interests.

=== Defamation ===
Article 40.3.2 states that “[t]he States shall . . . protect as best it may from unjust attack and, in the case of injustice done, vindicate the life, person, [and] good name . . . of every citizen.&lt;ref&gt;Irish Constitution, Article 40.3.2.&lt;/ref&gt;  Accordingly, the right to freedom of expression does not extend to unjust attacks on a citizen’s good name.  In an attempt to reconcile these two Constitutional provisions, the Irish parliament enacted the Defamation Act 1961, which was repealed and replaced by the Defamation Act 2009.&lt;ref&gt;Krizia Testa, Defending Speech: A Call for Irish Defamation Reform, Trinity College Law Review, https://trinitycollegelawreview.org/defending-speech-a-call-for-irish-defamation-reform/#:~:text=1.,expression%20is%20crucial%2C%20yet%20arduous.&lt;/ref&gt;  In addition to the relevant constitutional and statutory provisions, defamation law is also governed by common law and European human rights law.&lt;ref&gt;Paul Tweed, “Republic of Ireland Media Law Guide,” Carter-Ruck, https://www.carter-ruck.com/law-guides/defamation-and-privacy-law-in-republic-of-ireland/. “Article 10(2) of the European Convention on Human Rights . . . guarantees the ‘protection of the reputation . . . of others.’” ''Id''.&lt;/ref&gt;

The [https://www.irishstatutebook.ie/eli/2009/act/31/enacted/en/pdf Defamation Act of 2009] abolished the prior division between libel and slander and created one cause of action – the tort of defamation.&lt;ref&gt;''Id''.&lt;/ref&gt;  A cause of action for defamation exists where there is a “publication, by any means, to one or more than one person, of a defamatory statement, which ‘tends to injure a person’s reputation in the eyes of reasonable members of society.’”&lt;ref&gt;''Id''. (citing Defamation Act 2009).&lt;/ref&gt;  Defamation is governed only by civil law in Ireland, meaning that no criminal action can be taken for a violation of the law.&lt;ref&gt;''Id''.&lt;/ref&gt;

==== Defenses to Defamation ====
The law presumes that any defamatory statement is false, meaning that truth plays a vital role in defamatory law.&lt;ref&gt;''Id''.&lt;/ref&gt;  When a party alleges that they have been defamed, the burden is on the defendant to prove that the statement is true or demonstrate the existence of another defense.&lt;ref&gt;''Id''.&lt;/ref&gt;  If the statement is proven to be true or substantially true, the statement is not defamatory.&lt;ref&gt;Citizens Information, “The Law on Defamation in Ireland,” https://www.citizensinformation.ie/en/justice/civil-law/law-on-defamation/.&lt;/ref&gt;  This “reflects the principle that a person should not be compensated for damage done by a statement which is, in fact, true and accurate,” and the statement is essentially protected by the Constitution as an exercise of free expression and speech.&lt;ref&gt;Irish Legal Guide, “Defamation Defences,” https://legalguide.ie/defences/.&lt;/ref&gt;  A defendant may also have a defense of honest opinion, where the opinion is “honestly held . . . [meaning] the person making the statement believes the truth of the allegation at the time of making the statement” and related to a matter of public interest.&lt;ref&gt;Citizens Information, “The Law of Defamation in Ireland.”&lt;/ref&gt; ''' '''A defense of absolute privilege applies to certain defamatory statements “made in an official capacity or as part of a testimony.”&lt;ref&gt;''Id''.&lt;/ref&gt;  There is also a defense of qualified privilege, which applies to statements which the relevant persons had a duty or interest in making and receiving the statements.&lt;ref&gt;''Id''.&lt;/ref&gt;  Finally, as to publications, a court considers a number of factors in determining whether the publication was “fair and reasonable” or “innocent,” which have protected social media companies and newspapers from being liable for defamation.&lt;ref&gt;''Id''. These factors include the seriousness of the allegation, the content of the statement, and the relation of the statement to the public interest. ''Id''.&lt;/ref&gt;

==== Caselaw ====
In [https://ww2.courts.ie/ga/view/Judgments/959ff643-11b5-4e3d-89cc-5ae2a75f1232/043833d2-d57b-4967-a76f-4ded8cf42b82/2025_IEHC_90.pdf/pdf Stillorgan Gas Heating and Plumbing v. Manning], defamation law was applied to online comments, and the High court awarded Stillorgan €40,000 for comments that the defendants left on the company’s review pages.&lt;ref&gt;Lisa Carty &amp; Jane Bourke, “Recent Irish Defamation Cases Clarify Application of Law,” Pinsent Masons, (June 4 2024), https://www.pinsentmasons.com/out-law/analysis/recent-irish-defamation-cases-clarify-application-of-law.&lt;/ref&gt;

=== '''Terrorism''' ===
The European Union enacted Regulation 2021/784 to address “the dissemination of terrorist content online.”&lt;ref&gt;Official Journal of the European Union, Regulation (EU) 2021/74 of the European Parliament and of the Council, “On Addressing the Dissemination of Terrorist Content Online,” https://eur-lex.europa.eu/legal-content/EN/TXT/HTML/?uri=CELEX%3A32021R0784.&lt;/ref&gt;  In August 2025, the Irish Minister for Justice, Jim O’Callaghan, published the European Union (Online Dissemination of Terrorist Content), exercising the powers given to him by Section 3 of the European Communities Act 1972 in order to give further effect to Regulation (EU) 2021/784. &lt;ref&gt;Government of Ireland, “S.I. No. 375/2025 - European Union (Online Dissemination of Terrorist Content) (Designation of the Commissioner of An Garda Síochána as a Competent Authority) Regulations 2025,” (Aug. 1, 2025), https://www.irishstatutebook.ie/eli/2025/si/375/made/en/print.&lt;/ref&gt; The regulation requires “internet companies in the EU [to] take swift measures to prevent the misuse of their services for dissemination of terrorist content.”&lt;ref&gt;Thomas Wahl, “Rules on Removing Terrorist Content Online Now Applicable,” Eucrim, (June 7, 2022), https://eucrim.eu/news/rules-on-removing-terrorist-content-online-now-applicable/.&lt;/ref&gt;  

The framework set out in the regulation is directed toward precluding terrorists from using the internet to “recruit, encourage attacks, provide training and glorify their crimes.”&lt;ref&gt;''Id''.&lt;/ref&gt;  The primary components of the regulation include an “[o]bligation for Hosting Service Providers (HSPs) to remove terrorist content online within one hour after receiving a removal order from a competent national authority of an EU Member States; [l]imited scrutiny of cross-border removal orders by the competent authority of the Member State where the HSP has its main establishment or where its legal representative resides; [o]bligation for platforms to take proactive measures when they are exposed to terrorist content; [i]nclusion of several safeguards to ensure respect with fundamental rights, in particular freedom of expression and the right to information; [and an] [o]bligation for Member States to sanction platforms for non-compliance with the obligations under the Regulation.”&lt;ref&gt;''Id''.&lt;/ref&gt;  The Irish Minister for Justice, Home Affairs and Migration also published the 2025 Amendment to the Criminal Justice (Terrorist Offenses) Bill, giving full effect to EU Regulation 2017/541.&lt;ref&gt;Oireachtas, “Criminal Justice (Terrorist Offences) (Amendment) Bill 2025,” https://data.oireachtas.ie/ie/oireachtas/bill/2025/34/eng/initiated/b3425d.pdf.&lt;/ref&gt;  Regulation 2021 adopted the language defining terrorism from Regulation 2017/541 and applied it primarily to action relating to electronic communications and dissemination of information, as well as cyber-terrorism to address “the misuse of hosting services for terrorist purposes.”&lt;ref&gt;Joan Barata, “Regulation (EU) 2021/784 of the European Parliament and of the Council of 29 April 2021 on Addressing the Dissemination of Terrorist Content Online, Regulation on Terrorist Content Online,” (May 17, 2021), https://wilmap.stanford.edu/node/31158#:~:text=The%20Regulation%20establishes%20a%20definition,raising%20purposes%20against%20terrorist%20activity.&lt;/ref&gt;

=== '''Disinformation''' ===
There is not one existing regulatory framework in Ireland that specifically governs disinformation, but together, the regulations and existing media literacy policies do address disinformation.&lt;ref&gt;John Cian McGrath and Kirsty Park, Dublin City University, “EDMO Policy Monitoring: The Regulation of Online Disinformation,” (Nov., 2022), https://edmohub.ie/wp-content/uploads/2023/02/EDMO-Task-V-Dec-2022-2.pdf#:~:text=The%20Electoral%20Reform%20Act%20has%20been%20passed,not%20compatible%20with%20the%20e%2D%20Commerce%20Directive4.&lt;/ref&gt;  Specifically, four Irish regulations and policies relate to disinformation in the State: the Online Safety and Media Regulation Act, the Electoral Reform Act, the Broadcasting Authority of Ireland’s Media Literacy Policy, and The Future of Media Commission Report.&lt;ref&gt;''Id''.&lt;/ref&gt;

The Online Safety and Media Regulation Act has no particular aspect which regulates disinformation, but “tackles specific instances of online harms” and has the potential to include misinformation and disinformation through later amendments.&lt;ref&gt;''Id''.&lt;/ref&gt;  The Electoral Reform Act 2022 contains sections “which deal with regulating online advertising and online electoral information” and is aimed at protecting “the integrity of [Ireland’s] electoral and democratic process against the spread of disinformation and misinformation in the online sphere during electoral periods.”&lt;ref&gt;''Id''.&lt;/ref&gt;  The BAI Media Literary Policy facilitates Media Literacy Ireland (“MLI”), “an informal and voluntary alliance of organisations and individuals that aims to promote media literacy across the country.”&lt;ref&gt;''Id''.&lt;/ref&gt;  MLI provides training sessions and webinars aimed at tackling disinformation and helping consumers “tell the difference between reliable and accurate information and deliberately false or misleading information.”&lt;ref&gt;''Id''.&lt;/ref&gt;  MLI presents its work to the Irish Parliament.&lt;ref&gt;''Id''.&lt;/ref&gt;  The Future of Media Commission Report “addresses disinformation and recommends the development of a National Counter Disinformation Strategy” that would coordinate efforts against manipulation of Irish internet users via disinformation spread online.&lt;ref&gt;''Id''.&lt;/ref&gt;

== '''Cultural and Religious Expression''' ==
The legal framework governing cultural and religious expression in Ireland reflects a balance between constitutional freedoms and the limits on those freedoms for reasons of public order and morality.  The Irish Constitution guarantees freedom of expression and the right to assemble peacefully, rights which are commonly guaranteed by other nations to their citizens as well.  Cultural events and public festivals in Ireland must comply with a range of licensing and safety regulations, such as local authority permits and crowd control management.  The religious underpinnings in Ireland law can be seen in the nation’s previous blasphemy laws, which have since been abolished.

=== Public Festivals and Traditions ===
Article 40.6.1.ii of the Irish Constitution guarantees “the right of citizens to assemble peaceably and without arms.”&lt;ref&gt;Irish Constitution, Article 40.6.1.ii&lt;/ref&gt;  However, the right is “subject to public order and morality,” and may be limited in order to “prevent or control meetings which are determined in accordance with law to be calculated to cause a breach of the peace or to be a danger or nuisance to the general public or prevent or control meetings in the vicinity of either House of the Oireachtas.”&lt;ref&gt;''Id''.&lt;/ref&gt;  The Irish Constitution contains an explicit provision along with this guarantee which prohibits any law that regulates the right of free assembly from doing so on a political, religious, or class discrimination basis.&lt;ref&gt;Irish Constitution, Article 40.6.2.&lt;/ref&gt;

St. Patrick’s Day, which is observed on March 17th every year, originally began as a religious holiday, but has become a celebration of Irish culture over time.&lt;ref&gt;Rose Davidson, “St. Patrick’s Day,” National Geographic Kids, https://kids.nationalgeographic.com/celebrations/article/st-patricks-day.&lt;/ref&gt;  St. Patrick came to Ireland from Britain and is said to have “converted many of the country’s residents to Christians.”&lt;ref&gt;''Id''.&lt;/ref&gt;  Originally a very religious holiday to honor St. Patrick, churchgoers later began “celebrating their Irish heritage with cheeky pints of Guinness and live music . . . and it’s [now] more widely known as a global celebration of Irish culture.&lt;ref&gt;Kevin McGraw, “Surviving &amp; THriving: Tips for Celebrating St. Patrick’s Day in Ireland,” EF, (Feb. 2, 2024), https://www.efultimatebreak.com/blog/europe/ireland/tips-celebrating-st-patricks-day.&lt;/ref&gt;  The main event of the day in Ireland is the St. Patrick’s Day parade with many floats representing Irish folklore.&lt;ref&gt;''Id''.&lt;/ref&gt;

==== Regulation of Outdoor Events ====
The Planning and Development Act, 2000 established a system of licensing for outdoor events that apply to “new venues and events, which did not have specific planning permission prior to the 2000 Act.”&lt;ref&gt;S.I. No. 154/2001 - Planning and Development (Licensing of Outdoor Events) Regulations, 2001; Irish Legal Guide, “Outdoor Events,” Irish Law Explained, &lt;nowiki&gt;https://legalguide.ie/outdoor-events/&lt;/nowiki&gt;.&lt;/ref&gt;  Any event with an audience of 5,000 members or more is considered “prescribed” and requires a license.&lt;ref&gt;Irish Law Explained, &lt;nowiki&gt;https://legalguide.ie/outdoor-events/&lt;/nowiki&gt;.&lt;/ref&gt;  The Local Authority is responsible for reviewing the application, plans, and observations available for public inspection, and making a decision on the application.&lt;ref&gt;''Id''.&lt;/ref&gt;  The Local Authority may grant a license with or without conditions or outright refuse the license, and “may require compliance with guidance and codes of practice issued by the Minister.”&lt;ref&gt;''Id''.&lt;/ref&gt;  The Local Authority must consult with specific authorities, including the Garda police force, the Health Service Executive, and and any other local authorities particular to the area where the event is being held.&lt;ref&gt;''Id''.&lt;/ref&gt;  If a local authority believes that an event is occurring or likely to occur without a license, an enforcement notice may be served requiring the event or preparations of the event to immediately cease.&lt;ref&gt;''Id''.&lt;/ref&gt;  

In 2014, the licensing of events under the Planning and Development Act was amended after the issues regarding the proposed Garth Brooks concerts.&lt;ref&gt;''Id''.&lt;/ref&gt;  Brooks had five concerts scheduled in Dublin, but the Dublin city council denied permission for three of his concerts because of the noise, traffic, and potential behavior that a series of five concerts may create.  In response, Brooks canceled all five of the concerts he had planned.&lt;ref&gt;Sean Michaels, “Garth Brooks Calls Off Entire Run of Dublin Comeback Dates, (Jul. 9, 2014), &lt;nowiki&gt;https://www.theguardian.com/music/2014/jul/09/garth-brooks-cancels-dublin-comeback-dates&lt;/nowiki&gt;.&lt;/ref&gt;  The amendments made is mandatory for event organizers which planned to make an application to consult with the local “in order to discuss the submission of an application, including the draft plan for the management of the event.”&lt;ref&gt;S.I. No. 154/2001 - Planning and Development (Licensing of Outdoor Events) Regulations, 2001.&lt;/ref&gt;

==== Crowd Control ====
Crowd control regulations are used at public events in Ireland “to maintain public peace and order and ensure the safety of all who are gathered.”&lt;ref&gt;Citizens Information, “Crowd Control at Public Events, https://www.citizensinformation.ie/en/justice/law-enforcement/crowd-control-at-public-events-in-ireland/.&lt;/ref&gt;  The Garda Síochána, the Irish police force, is given legal authority to manage crowd control by the Criminal Justice (Public Order) Act 1994.&lt;ref&gt;''Id''.&lt;/ref&gt;  The police force is empowered to place barriers on roads up to one mile from where a large public event is taking place and may seize alcohol or other drink containers where it deems necessary.&lt;ref&gt;''Id''.&lt;/ref&gt;  The police force also may prevent public events from taking place within a half mile of the House of the Oireachtas.&lt;ref&gt;''Id''.&lt;/ref&gt;

=== Religious Discrimination ===
Article 44.2.1 guarantees Irish citizens the &quot;free profession and practice of religion . . . subject to public order and morality.&lt;ref&gt;Irish Constitution, Article 44.2.1.&lt;/ref&gt;  The Article also provides that the State will not &quot;endow any religion,&quot; nor &quot;impose any disabilities or make any discrimination on the ground of religious profession, belief of status.&quot;&lt;ref&gt;Irish Constitution, Article 44.2.2-3.&lt;/ref&gt;  Article 44, therefore, provides for freedom of religion and implicitly protects individuals from religious discrimination.&lt;ref&gt;&quot;2023 Report on International Religious Freedom: Ireland,&quot; U.S. Department of State, https://www.state.gov/reports/2023-report-on-international-religious-freedom/ireland/.&lt;/ref&gt;  In 1972, an amendment was made to the Constitution which removed the recognition of the Catholic Church and other specific denominations.&lt;ref&gt;''Anti-Discrimination Legislation'', Civic Nation (last visited Dec. 1, 2025), https://civic-nation.org/ireland/government/legislation/anti-discrimination_legislation/.&lt;/ref&gt;  

The clauses have since been compared to those protections which are often enjoyed by citizens of countries with a secular constitution, such as those guaranteed by the First Amendment of the U.S. Constitution.&lt;ref&gt;David Kenny, ''God in the Irish Constitution'', BYU Law (Oct. 31, 2020), https://talkabout.iclrs.org/2020/10/31/god-in-the-irish-constitution/.&lt;/ref&gt; Still, the influence of God can be seen in the way that Irish courts have interpreted and applied the constitutional provisions.&lt;ref&gt;''Id''.&lt;/ref&gt;  In the 1972 case of ''Quinn Supermarkets v. Attorney General'', the Irish Supreme Court found that “where discrimination in favour of religion is necessary to enable freedom of religious practice,” the Constitution would allow and possible require such discrimination.&lt;ref&gt;''Id''.&lt;/ref&gt;  The Court therefore held that the main goal of the Constitution’s non-discrimination clause was to protect the free practice of religion.&lt;ref&gt;''Id''.&lt;/ref&gt;  Similarly, the 1979 case of ''McGrath and Ó Ruairc v. Trustees of Maynooth College'', two priests had been discharged from the religious educational institution where they taught.&lt;ref&gt;''Id''.&lt;/ref&gt;  The priests brought a discrimination claim, but lost at the Supreme Court, which found that &quot;the rights of the religious college not only allowed but required this discrimination.&quot;&lt;ref&gt;''Id''.&lt;/ref&gt;  

Article 44 of the Irish Constitution further provides that “[l]egislation providing State aid for  schools shall not discriminate between schools under the management of different religious denominations, nor be such as to affect prejudicially the right of any child to attend a school receiving public money without attending religious instruction at that school.”&lt;ref&gt;Irish Constitution, Article 44.2.4.&lt;/ref&gt;  In Ireland, &quot;national schools&quot; are privately owned and managed, but are funded by the State.&lt;ref&gt;&quot;2023 Report on International Religious Freedom: Ireland,&quot; U.S. Department of State, https://www.state.gov/reports/2023-report-on-international-religious-freedom/ireland/.&lt;/ref&gt;  The majority of the national schools are affiliated with a religious group, 88% of which are affiliated with the Roman Catholic Church.&lt;ref&gt;''Id''.&lt;/ref&gt;. Irish courts have been lenient in enforcing the endowment provisions and allow state funding for religious chaplains in school.&lt;ref&gt;David Kenny, ''God in the Irish Constitution'', BYU Law (Oct. 31, 2020), https://talkabout.iclrs.org/2020/10/31/god-in-the-irish-constitution/.&lt;/ref&gt;  Previously, state-funded schools that were affiliated with a religion were allowed to favor admitting children that were members of the relevant religious group, resulting in discrimination to non-religious children or those that were members of a different religion.&lt;ref&gt;''Id''.&lt;/ref&gt;   

=== Blasphemy Laws ===
Given that much of Ireland’s legal system was influenced by England’s common law system, Irish blasphemy law was similarly shaped by English blasphemy jurisprudence.&lt;ref&gt;Seosamh Gráinséir, ''Irish Legal Heritage: Blasphemy Law in Ireland'', Irish Legal News (Oct. 19, 2018), https://www.irishlegal.com/articles/irish-legal-heritage-blasphemy-law-in-ireland.&lt;/ref&gt;  Therefore, when adopted in 1937, the Irish Constitution explicitly included a prohibition on blasphemy as a limit to the right to freedom of speech, making the “publication or utterance of blasphemous, seditious, or indecent matter [ ] an offense which shall be punishable in accordance with law.”&lt;ref&gt;Irish Constitution, Article 40.6.1.i.&lt;/ref&gt;

In 2009, the Irish parliament passed the Defamation Act, which provided that “[a] person who publishes or utters blasphemous matter shall be guilty of an offence and shall be liable upon conviction on indictment to a fine not exceeding €25,000.”&lt;ref&gt;Defamation Act 2009, Part 5.36.&lt;/ref&gt;  The Act added clarity to what constituted blasphemous materials, describing such material as “matter that is grossly abusive or insulting in relation to matters held sacred by any religion, thereby causing outrage among a substantial number of the adherents of that religion,” but that a person must intend to cause such outrage to be found guilty of the offense.&lt;ref&gt;''Id''.&lt;/ref&gt;

Campaigns to abolish the &quot;medieval&quot; ban on blasphemy spurred since the law’s enactment in 2009.&lt;ref&gt;Emma Graham-Harrison, ''Ireland Votes to Oust ‘Midieval’ Blasphemy Law'', The Guardian (Oct. 27, 2018), https://www.theguardian.com/world/2018/oct/27/ireland-votes-to-oust-blasphemy-ban-from-constitution.&lt;/ref&gt;  In 2018, a referendum allowed citizens to vote on whether to remove the offense from the Constitution.&lt;ref&gt;''Id''.&lt;/ref&gt;  Approximately 65% of the voters wanted to remove the prohibition on blasphemy, although polls show that only 45% of eligible voters actually cast a ballot.&lt;ref&gt;''Id''.&lt;/ref&gt;  Following the vote, amendments were made to the Constitution and Defamation Act to remove the relevant clauses.&lt;ref&gt;''Irish Vote to Scrap Offence of Blasphemy'', British Broadcasting Corporation (Oct. 28, 2028), https://www.bbc.com/news/world-europe-46010077#.&lt;/ref&gt;  The Justice Minister at the time agreed with the result, stating that there was “no room for a provision such as this in [the] Constitution.”&lt;ref&gt;''Id''.&lt;/ref&gt;

===== Caselaw =====
The earliest case of blasphemy reported in Ireland was the trial of Thomas Emlyn, a unitarian minister.&lt;ref&gt;Seosamh Gráinséir, ''Irish Legal Heritage: Blasphemy Law in Ireland'', Irish Legal News (Oct. 19, 2018), https://www.irishlegal.com/articles/irish-legal-heritage-blasphemy-law-in-ireland.&lt;/ref&gt;  He was punished for publishing ''An Humble Inquiry into the Scripture-Account of Jesus Christ'', which “argued that Jesus Christ was not the equal of God.”&lt;ref&gt;''Id''.&lt;/ref&gt;  Emlyn was found guilty of blasphemy and sentenced to one year in prison and ordered to pay a £1000 fine.&lt;ref&gt;''Id''.&lt;/ref&gt; Around 1995, as new divorce legislation was being voted on in Ireland, various publications were marked as containing blasphemous material through “their depiction of the progressive separation of church influence from the governance of Ireland,&quot; but there were no resulting prosecutions.&lt;ref&gt;Seosamh Gráinséir, ''Irish Legal Heritage: Blasphemy Law in Ireland'', Irish Legal News (Oct. 19, 2018), &lt;nowiki&gt;https://www.irishlegal.com/articles/irish-legal-heritage-blasphemy-law-in-ireland&lt;/nowiki&gt;&lt;/ref&gt;  

In 1999, the Supreme Court of Ireland held in ''Corway v. Independent Newspaper'' that a cartoon which included the phrase “Hello progress – bye bye Father?” was “in very bad taste,” but refused to prosecute because there was no clear legislative definition of criminal blasphemy.&lt;ref&gt;Seosamh Gráinséir, ''Irish Legal Heritage: Blasphemy Law in Ireland'', Irish Legal News (Oct. 19, 2018), https://www.irishlegal.com/articles/irish-legal-heritage-blasphemy-law-in-ireland.&lt;/ref&gt;  Although the Defamation Act 2009 did remedy this issue, creating a specific offense for blasphemy, there were no successful prosecutions brought under the Act.&lt;ref&gt;''Id''.&lt;/ref&gt;  However, prior to the calls for its abolishment, the laws made headlines in 2017 when an investigation into Stephen Fry, a British comedian and actor, was announced for his 2015 interview on Ireland’s public service broadcaster.&lt;ref&gt;''Ireland'', End Blasphemy Laws (last updated Jul. 14, 2020), https://end-blasphemy-laws.org/countries/europe/ireland/.&lt;/ref&gt;  During the interview, Fry referred to God as “a capricious, mean-minded, stupid[,] . . . maniac.”&lt;ref&gt;''Id''.&lt;/ref&gt;  The Irish police’s investigation did not result in any criminal liability for Fry because there was not enough &quot;public outrage&quot; over the incident, as campaigns for the abolishment of blasphemy laws were already underway.&lt;ref&gt;''Id''.&lt;/ref&gt;

== '''Privacy and Data Protection''' ==
Privacy and data protection law in Ireland is centered around the protection of an individuals' personal autonomy and control over their personal information, while ensuring the needs of relevant authorities are met where necessary.  The Irish Constitution guarantees the right of privacy, which has been developers through judicial interpretation and expanded through data protection laws.  Largely shaped by EU law, the GDPR is enforced at the national level and provides guidelines and regulations which Ireland must implement with respect to data protection.  Together, the regulatory framework seeks to uphold the right of privacy while balancing freedom of expression and individual interests.

=== Individual Right to Privacy ===
Under the common law in Ireland, there was no specific tort action which protected an individual’s privacy.&lt;ref&gt;Irish Legal Guide, “Privacy Rights,” [https://legalguide.ie/privacy-rights-2/. https://legalguide.ie/privacy-rights-2/.]&lt;/ref&gt;  The Irish Constitution grants citizens an implied right to privacy, but very few cases have addressed the right to privacy.&lt;ref&gt;''Id''.&lt;/ref&gt;  “The European Convention on Human Rights provides that everyone has the right to respect for his private and family life, his home and his correspondence.”&lt;ref&gt;''Id''.&lt;/ref&gt;  Under the ECHR, “[t]here shall be no interference with the right to privacy family life and the home from unwarranted interference by the state.”&lt;ref&gt;''Id''.&lt;/ref&gt;

==== Irish Constitution ====
Irish courts have acknowledged that the Constitution affords a right to privacy and anonymity, but that right is not unqualified and applies only in limited circumstances.&lt;ref&gt;''Id''.&lt;/ref&gt;  The right is recognized as an implied right granted under Article 40.3.1 of the Irish Constitution.&lt;ref&gt;''Id''.&lt;/ref&gt;  In the ''McGee'' case, the Irish Supreme Court the right to marital privacy was recognized as “protected by the Constitution” that “inheres in an individual, by reason of their human personality.”&lt;ref&gt;''Id''.&lt;/ref&gt;  The right cannot be intruded on by the State “without good objective justification, in relation to decisions taken within the scope of the zone of privacy.”&lt;ref&gt;''Id''.&lt;/ref&gt;  In the case of ''Kennedy and others v. Ireland'', the illegal phone tapping by Irish officials was determined to have breached the right to privacy for a journalist.&lt;ref&gt;''Id''.&lt;/ref&gt;

In Ireland, the right to privacy can be limited in instances where it would be outweighed by the common good, public interest, or other Constitutional freedoms.&lt;ref&gt;''Id''.&lt;/ref&gt;  In cases that relate to media communication, the courts have afforded the right to freedom of expression more protection than the right to privacy.&lt;ref&gt;''Id''.&lt;/ref&gt;

==== Regional Law ====
Irish law has adopted the European Convention on Human Rights Act 2003, which provides that “everyone has the right to respect for his private and family life, his home and correspondence [and] that there should be no interference by a public authority with the exercise of this right, except in accordance with the law where it is necessary in a democratic society in the interests of national security, public safety or the economic wellbeing of the country, for the prevention of disorder or crime or for the protection of health or morals, rights and freedom of others.”&lt;ref&gt;''Id''.&lt;/ref&gt;

Case law interpreting the scope of the right to privacy under the Convention have found that “privacy may encompass a person’s personal information, genetic information, photograph, identity, reputation and honour.&lt;ref&gt;''Id''.&lt;/ref&gt;

=== Data Protection ===
The primary data protection laws in Ireland are the General Data Protection Regulation (“GDPR”) and the Data Protection Act 2018 (“Irish DPA”).&lt;ref&gt;“Frequently Asked Questions about Data Protection and Privacy Rights (under GDPR),” Adoption Authority of Ireland, https://aai.gov.ie/en/who-we-are/data-protection-gdpr/faqs-data-protection-gdpr.html#:~:text=1.,with%20our%20data%20protection%20obligations.&lt;/ref&gt;  Both of these laws address the protections afforded to individuals in Ireland regarding the processing of personal data.&lt;ref&gt;''Id''.&lt;/ref&gt;  The laws impose various requirements on controllers and processes of personal data and guarantee individuals with specific rights related to their data.&lt;ref&gt;''Id''.&lt;/ref&gt;

==== Legislation ====
The European Union General Data Protection Regulation (Regulation (EU) 2016/679) (“GDR”), directly applicable to all EU Member States.&lt;ref&gt;Irish Legal Guide, “Privacy Rights,” [https://legalguide.ie/privacy-rights-2/. https://legalguide.ie/privacy-rights-2/.]&lt;/ref&gt;

In Ireland, the Data Protection Act provides a higher degree of protection to sensitive personal data.  Sensitive and personal data includes information “relating to an individual’s racial or ethnic origin, political opinions, religious or philosophical beliefs, physical or mental health, sexual life, the allegations or the commission of an offence.”&lt;ref&gt;''Id''.&lt;/ref&gt;  Exceptions under the Data Protection Act exist for certain purposes, such as journalism, art, and literary needs.&lt;ref&gt;''Id''.&lt;/ref&gt;

The GDPR is an EU data protection law which Ireland has adopted.&lt;ref&gt;“Your Rights Under the GDPR,” Data Protection Commission, [https://www.dataprotection.ie/en/individuals/rights-individuals-under-general-data-protection-regulation#:~:text=The%20GDPR%20also%20states%20that:%20*%20Data,feel%20their%20rights%20are%20not%20being%20respected. https://www.dataprotection.ie/en/individuals/rights-individuals-under-general-data-protection-regulation#:~:text=The%20GDPR%20also%20states%20that:%20*%20Data,feel%20their%20rights%20are%20not%20being%20respected.]&lt;/ref&gt;  The rights, as set out in Article 8 of the EU Charter of Fundamental Rights states that: “[E]veryone has the right to the protection of personal data concerning him or her[; s]uch data must be processed fairly for specified purposes and on the basis of the consent of the person concerned, or some other legitimate basis laid down by law[; e]veryone has the right of access to data which has been collected concerning him or her, and the right to have it rectified[; c]ompliance with these rules shall be subject to control by an independent authority.”&lt;ref&gt;''Id''.&lt;/ref&gt;  Therefore, “in order to process personal data, organizations must have a lawful reason.”&lt;ref&gt;''Id''.&lt;/ref&gt;  The GDPR recognizes six lawful reasons: consent; to carry out a contract; to meet a legal obligation; where the personal data is required to protect the vital interests of a person; where the personal data is required for a task to be carried out in the public interest; and in the legitimate interests of a company or organization unless those interests would harm an individual’s rights or freedoms. &lt;ref&gt;''Id''.&lt;/ref&gt;

The Irish DPA integrated the GDPR and transposes the EU Directive in Ireland, incorporating most of the GDPR provisions with certain additions and deletions in accordance with Irish law.&lt;ref&gt;Maria Khan, “What is Irish Data Protection Act of 2018, Securiti, (last updated Aug. 22, 2024), [https://securiti.ai/what-is-irish-data-protection-act-of-2018/#:~:text=The%20Irish%20DPA%20also%20outlines%20the%20responsibilities,data%20protection%20and%20enforcing%20GDPR%20in%20Ireland. https://securiti.ai/what-is-irish-data-protection-act-of-2018/#:~:text=The%20Irish%20DPA%20also%20outlines%20the%20responsibilities,data%20protection%20and%20enforcing%20GDPR%20in%20Ireland.]&lt;/ref&gt;  The Irish DPA affords the same rights to individuals regarding their personal data as that of the GDPR.&lt;ref&gt;''Id''.&lt;/ref&gt;  These rights include the right to be informed when and how an individual’s personal data is being used and collected, the right to access personal data, the right to restriction of processing data, the right to data portability across IT environments, the right to object to personal data being used for certain purposes, the right to not be discriminated against, and the right to erasure.&lt;ref&gt;''Id''.&lt;/ref&gt;

==== Children's Personal Data ====
Children are given the same data protection rights as adults, but have special protection of their personal data.&lt;ref&gt;Overview of the General Data Protection Regulation (GDPR), Citizens Information, (last updated Feb. 17, 2023), https://www.citizensinformation.ie/en/government-in-ireland/data-protection/overview-of-general-data-protection-regulation/.&lt;/ref&gt;  This is because children “may be less aware of the risks and consequences of sharing their personal data” and “less aware of the safeguards available and their rights in relation to how personal information is processed.”&lt;ref&gt;''Id''.&lt;/ref&gt;  Parents and guardians have the ability to make access requests or exercise other data protection rights on behalf of their children, but it is ultimately left to the data controller to determine what is in the child’s best interest in such circumstances. &lt;ref&gt;''Id''.&lt;/ref&gt;

Online service providers, such as social media companies, are able to rely on a child’s own consent for processing their personal data at age 16, as established by the Data Protection Act 2018.  Otherwise, a child’s parent or legal guardian must give consent.&lt;ref&gt;''Id''.&lt;/ref&gt;

==== Regulatory Body ====
The Data Protection Commission (“DPC”) is the organization “responsible for upholding the fundamental rights of individuals in the European Union to have their personal data protected.”&lt;ref&gt;''Id''.&lt;/ref&gt;  The Commission observes the relevant organizations to ensure that they are complying with the GDPR and other data protection laws and addresses complaints related to the breach of data protection rights.&lt;ref&gt;''Id''.&lt;/ref&gt;

== '''Right to Bodily, Spiritual and Digital Identity''' ==
Irish law recognized bodily integrity, spiritual identity, and digital identity as fundamental rights.  These are primarily protected under the Irish Constitution and expanded by judicial interpretation. The areas have been further developed under legislation and caselaw with specific respect to gender recognition, birth information, and digital privacy. Influenced by EU law and the ECHR, Irish law works to balance individual interest and self-determination with the interests of the State.

=== Right to Bodily Integrity ===
Article 40.3.1 of the Irish Constitution guarantees individuals “laws to respect, and, as far as practicable, . . . laws to defend and vindicate the personal rights of the citizen.”&lt;ref&gt;&quot;Article 40.3.1, The Right to Bodily Integrity,&quot; Irish Council for Human Rights, [https://ichr.ie/our-work/#:~:text=The%20ICHR%20shall%20work%20tirelessly,beings%20over%20their%20own%20bodies. https://ichr.ie/our-work/#:~:text=The%20ICHR%20shall%20work%20tirelessly,beings%20over%20their%20own%20bodies.]&lt;/ref&gt;  The right to bodily integrity has been recognized as an unenumerated right under this Article.&lt;ref&gt;''Id''.&lt;/ref&gt;  In the case of ''Gladys Ryan v. The Attorney General'', the right to bodily integrity was interpreted “to mean that no mutilation of the body or any of its members may be carried out on any citizen under any authority of the law except for the good of the whole body and that no process which is or may, as a matter of probability, be dangerous or harmful to the life or health of the citizens or any of them may be imposed . . . by an Act of the Oirechtas.”&lt;ref&gt;''Id''.&lt;/ref&gt;  The right relates to the protection of “personal autonomy, self-ownership, and self-determination of human beings over their own bodies.”&lt;ref&gt;''Id''.&lt;/ref&gt;

=== Personal Identity Law ===
Article 12 of the Universal Declaration of Human Rights and Article 8 of the European Convention on Human Rights provide the right to one’s identity.&lt;ref&gt;''A Case Study of the Right to One’s Identity in Relation to Ireland’s History of the Institutionalisation of Citizens in the Current Context of the Birth Information and Tracing Act 2022'', Irish Centre for Human Rights (Feb. 16, 2023), [https://ichrgalway.org/2023/02/16/a-case-study-of-the-right-to-ones-identity-in-relation-to-irelands-history-of-the-institutionalisation-of-citizens-in-the-current-context-of-the-birth-information-and-tracing-act-2022/#:~:text=The%20Birth%20Information%20and%20Tracing%20Act%202022%20firmly%20and%20finally,the%20Convention%27%20as%20per%20Art. https://ichrgalway.org/2023/02/16/a-case-study-of-the-right-to-ones-identity-in-relation-to-irelands-history-of-the-institutionalisation-of-citizens-in-the-current-context-of-the-birth-information-and-tracing-act-2022/#:~:text=The%20Birth%20Information%20and%20Tracing%20Act%202022%20firmly%20and%20finally,the%20Convention%27%20as%20per%20Art.]&lt;/ref&gt;  The ECHR states that “everyone has the right to respect for his private and family life, his home and his correspondence.”&lt;ref&gt;''Id''.&lt;/ref&gt;  This article has been interpreted to provide the right to one’s identity.&lt;ref&gt;''Id''.&lt;/ref&gt; As an EU member bound by te ECHR, Ireland is “required to ‘secuer to everyone within their jurisdiction the rights and freedoms defined” in the ECHR, which includes the right to one’s identity.&lt;ref&gt;''Id''.&lt;/ref&gt;

The first instance when the right to identity was recognized in Ireland in relation to adoption was in the case of ''O’T v. B'' in 1998.&lt;ref&gt;Jamie Aspell, ''Ireland’s Birth Information and Tracing Act: Reconciling the Right to Identity'', Blog of the European Journal of International Law (Sep. 23, 2022), https://www.ejiltalk.org/irelands-birth-information-and-tracing-act-reconciling-the-right-to-identity/.&lt;/ref&gt;  The Irish court held that “the right to identity was not absolute and had to be balanced against the mother’s right to privacy.”&lt;ref&gt;''Id''.&lt;/ref&gt;  The case was commonly cited for the proposition that adoptees did not have unequivocal access to their birth information.&lt;ref&gt;''Id''.&lt;/ref&gt;  A decade later, in the 2008 case of ''South Western Area Health Board v. Information Commissioner'', the Irish High Court found that “the disclosure of records containing an adopted person’s birth information was impermissible, as the birth mother had not been accorded an opportunity to ‘make representations in support of the rights she sought to protect.’”&lt;ref&gt;''Id''.&lt;/ref&gt;  The cases led to confusion over how the constitutional right to privacy and the constitutional right to one’s identity can operate with one another in the adoption context.&lt;ref&gt;''Id''.&lt;/ref&gt;

==== Birth Information and Tracing Act 2022 ====
In June 2022, Ireland enacted the [https://www.irishstatutebook.ie/eli/2022/act/14/enacted/en/html Birth Information and Tracing Act].&lt;ref&gt;Birth Information and Tracing Act, 2022.&lt;/ref&gt;  This law established a “right to full birth, early life, care and medical information for all those with questions on their origins.”&lt;ref&gt;''Id''.&lt;/ref&gt;  The law allows people who were adopted to access information about their birth parents’ identity once they reach the age of 16.&lt;ref&gt;''Id''.&lt;/ref&gt;  The intent behind the legislation was to right previous wrongs that had been committed to adopted people in Ireland. &lt;ref&gt;Jamie Aspell, ''Ireland’s Birth Information and Tracing Act: Reconciling the Right to Identity'', Blog of the European Journal of International Law (Sep. 23, 2022), https://www.ejiltalk.org/irelands-birth-information-and-tracing-act-reconciling-the-right-to-identity/.&lt;/ref&gt; Specifically, this was seen as a response to the Irish Government’s commission of the Mother and Baby Homes Report, which “investigated the abuses perpetrated against women and children in institutions which were used throughout 20th century Ireland to house women who became pregnant outside of marriage.”&lt;ref&gt;''Id''.&lt;/ref&gt;  The report recommended that adopted persons from such institutions should have an established right to their birth information.&lt;ref&gt;''Id''.&lt;/ref&gt;  This report and recommendation is viewed as a “catalyst for the Birth Information Tracing Act.”&lt;ref&gt;''Id''.&lt;/ref&gt;
The Act seems to lend support to the idea that Ireland is prioritizing the right to identity over the privacy of parents in the context of adoption.&lt;ref&gt;''Id''.&lt;/ref&gt;  Other EU member countries have taken different approaches, such as France, which allows mothers of adoptees to waive their anonymity.&lt;ref&gt;''Id''.&lt;/ref&gt;  While the ECtHR has not expressly taken a position as to which of these rights should be prioritized, case law seems to suggest that the ECtHR has &quot;tended to prioritise the right to identity over the privacy rights of parents,” similar to the legislation in Ireland.&lt;ref&gt;''Id''.&lt;/ref&gt;  This can be seen in ''Odièvre v. France'', in which the ECtHR decided a case regarding the practice of anonymous birthing, in which a mother can give up her newborn child for adoption and request that her identity remain anonymous, as protected by the French Civil Code.&lt;ref&gt;''Id''.&lt;/ref&gt;  The Court’s holding suggested that there should be a balance in legislation that ensures protection of the privacy rights of a mother and competing rights to identity of an adopted child.&lt;ref&gt;''Id''.&lt;/ref&gt;

==== Gender Recognition Act of 2015 ====
The Gender Recognition Act (“GRA”) of 2015 established the rights of transgender individuals in Ireland.&lt;ref&gt;''The Role of Self-Determination in Ireland’s Gender Recognition Act'', Washington Center for Human Rights (Oct, 22, 2025), [https://washingtoncentre.org/the-role-of-self-determination-in-irelands-gender-recognition-act/#:~:text=The%20Gender%20Recognition%20Act%20(GRA,gender%20identity%20of%20medical%20necessity. https://washingtoncentre.org/the-role-of-self-determination-in-irelands-gender-recognition-act/#:~:text=The%20Gender%20Recognition%20Act%20(GRA,gender%20identity%20of%20medical%20necessity.]&lt;/ref&gt;  Prior law in Ireland was “based on medical gatekeeping to legitimize the identity of transgender.”&lt;ref&gt;''Id''.&lt;/ref&gt;  The 2015 legislation “provides a process enabling trans people to achieve full legal recognition of their preferred gender and allows for the acquisition of a new birth certificate that reflects this change.”&lt;ref&gt;''Id''.&lt;/ref&gt;  The law applies to individuals over 18, who can declare their own gender identity, and allows for people aged 16-17 to be recognized as their declared gender under another process.&lt;ref&gt;''Id''.&lt;/ref&gt;

When the GRA was enacted, Ireland was the last EU country that did not allow for the legal recognition of transfer people.&lt;ref&gt;''Id''.&lt;/ref&gt;  The process leading to the GRA took over five years, with various published frameworks and debate until the GRA was accepted and enacted.&lt;ref&gt;''Id''.&lt;/ref&gt;  The law came after Thomas Hammarberg, the European Commission for Human Rights, announced the need for EU members to adopt their own system for recognizing the preferred gender of their citizens.&lt;ref&gt;''Id''.&lt;/ref&gt;  Additional international pressure came from the UN Human Rights Committee’s report regarding Ireland’s conformity with the ICCPR.&lt;ref&gt;''Id''.&lt;/ref&gt;  The Committee specifically raised concerns with the need for a gender recognition bill.&lt;ref&gt;''Id''.&lt;/ref&gt;  

Since its enactment, “more than 70 percent of respondents show their support to transgender rights, which is a sharp contrast to the attitude of the period before 2015.”&lt;ref&gt;''Id''.&lt;/ref&gt;  A large cultural shift has emerged supporting inclusivity in school programs and national campaigns.&lt;ref&gt;''Id''.&lt;/ref&gt;  The Irish Department of Education has introduced new gender-neutral policies for school uniform and paperwork.&lt;ref&gt;''Id''.&lt;/ref&gt;  The current point of contention with regard to gender recognition relates to legislation allowing the formal recognition of non-binary and intersex people, which was not included in the GRA when enacted.&lt;ref&gt;''Id''.&lt;/ref&gt;

=== Digital Identity ===
As the digital services offered by the private and public spheres have grown, the need for secure digital authentication of individuals’ identities has also grown.&lt;ref&gt;“European Digital Identity,” Digital Economy and Society, Directorate-General for Communication of the European Commission, [https://commission.europa.eu/topics/digital-economy-and-society/european-digital-identity_en#:~:text=Digital%20Identity%20for%20all%20Europeans,by%20the%20end%20of%202026. https://commission.europa.eu/topics/digital-economy-and-society/european-digital-identity_en#:~:text=Digital%20Identity%20for%20all%20Europeans,by%20the%20end%20of%202026.]&lt;/ref&gt;  Threats to digital privacy have also emerged and individuals are concerned about profiling and surveillance. &lt;ref&gt;''Id''.&lt;/ref&gt; The EU has created the EU Digital Identity Framework in response.&lt;ref&gt;''Id''.&lt;/ref&gt;  The Framework is “based on the principle that everyone should always control their digital identity.”&lt;ref&gt;''Id''.&lt;/ref&gt;  The Framework involved EU Digital Identity Wallets (“eID wallets”) which would allow individuals to carry a digital wallet with them across all of the EU without any issues across borders or worries about losing control of personal data, while prioritizing privacy and security.&lt;ref&gt;''Id''.&lt;/ref&gt;  The eID wallets would “enable users to access online and offline public and private services, store and share digital documents, and create binding signatures.”&lt;ref&gt;''Id''.&lt;/ref&gt;  As a member state, Ireland would have to make eID wallets available to its citizens, residents, and businesses by the end of 2026.&lt;ref&gt;''Id''.&lt;/ref&gt;

== '''Right to Reject Information, Clothing and Human Exhibitions''' ==
In Ireland, the right to erasure laws are governed by the GDPR, which seeks to balance the right of one's access to their personal data and freedom of expression with the public interest. By contrast, laws relating to nudity and child pornography are rooted in criminal and public order laws, regulating public behavior to protect morality.  Clothing is not heavily regulated in Ireland and allows for the wearing of religious and cultural clothing as a form of freedom of expression and religion.

=== Right to Object/Erasure ===
The right to erasure, also known as the “right to be forgotten” is given by Articles 17 and 19 of the GDPR.&lt;ref&gt;''The Right to Erasure (Articles 17 and 19 of the GDPR)'', Data Protection Commission (last visited Dec. 3, 2025), [https://www.dataprotection.ie/en/individuals/know-your-rights/right-erasure-articles-17-19-gdpr. https://www.dataprotection.ie/en/individuals/know-your-rights/right-erasure-articles-17-19-gdpr.]&lt;/ref&gt;  The articles give Irish citizens “the right to have [their] data erased, without undue delay, by the data controller” in the following circumstances: (1) where personal data is “no longer necessary in relation to the purpose for which it was collected or processed;” (2) where consent is withdrawn and there is no other lawful basis for retaining the data; (3) where a person objects to the processing of their data and there is no legitimate grounds for continuing; (4) where a person objects and their personal data is being used for marketing purposes; (5) where personal data has been processed unlawfully; (6) where personal data has been erased for legal reasons; and (7) where personal data has been collected in relation to offering information to a child.&lt;ref&gt;''Id''.&lt;/ref&gt;  The right does not apply where the processing of personal data is “necessary for exercising the right of freedom of expression and information,” to comply with legal obligations, if necessary to preserve public interest, or if being used to establish a legal defense.&lt;ref&gt;''Id''.&lt;/ref&gt;  The right is also limited under Sections 43 and 60 of the Data Protection Act, which provide for the right of freedom of expression and information and restrictions necessary for the public interest.&lt;ref&gt;''Id''.&lt;/ref&gt;

=== Cookie Consent ===
In 2019, the DPC sent out a questionnaire to relevant organizations to examine the use of cookies across popular Irish websites.&lt;ref&gt;''Irish DPC Publishes New Cookie Guidance'', Hunton (Apr. 8, 2020), &lt;nowiki&gt;https://www.hunton.com/privacy-and-information-security-law/irish-dpc-publishes-new-cookie-guidance#:~:text=If%20a%20cookie%20is%20used,actual%20facts%20of%20the%20processing&lt;/nowiki&gt;.&lt;/ref&gt;  The goal of DPC’s sweep was “to examine how cookies and similar technologies are deployed, and to establish how and whether organizations are (1) complying with the current Irish cookie law rules, implementing the EU ePrivacy Directive into Irish law (the “Irish ePrivacy Regulations”) in particular; and (2) whether users’ consent for non-necessary cookies or tracking technologies is being obtained in line with the requirements of the [GDPR].”&lt;ref&gt;''Id''.&lt;/ref&gt;  The DPC offered new cookie guidance in which cookie consent has a lifespan of six months, like the CNIL in France, and must then be renewed.&lt;ref&gt;''Id''.&lt;/ref&gt;

In 2024, the DPC announced its decision after an investigation into LinkIn for its alleged use of personal data for behavioral analysis and targeted advertising.&lt;ref&gt;''Irish Data Protection Commission Fines LinkedIn Ireland €310 million'', Data Protection Commission (Oct. 24, 2024), &lt;nowiki&gt;https://www.dataprotection.ie/en/news-media/press-releases/irish-data-protection-commission-fines-linkedin-ireland-eu310-million&lt;/nowiki&gt;.&lt;/ref&gt;  The DPC determined that LinedIn had used personal data in violation of multiple Articles of the GDPR and fined the company €310 million, along with a reprimand and order to bring its processing in compliance with the GDPR.&lt;ref&gt;''Id''.&lt;/ref&gt;

=== Right to Clothing ===
&lt;u&gt;Textile Labeling:&lt;/u&gt; As an EU member, the EU Textile Fibre Names and Related Labelling and Marking of the Fibre Composition of Textile Products Regulations 2012 (S.I. No. 142 of 2012) governs Irish textiles.&lt;ref&gt;''Textile Labelling Regulations'', Competition and Consumer Protection Commission (last updated Dec. 5, 2025), &lt;nowiki&gt;https://www.ccpc.ie/business/help-for-business/guidelines-for-business/textile-labelling-regulations/&lt;/nowiki&gt;.&lt;/ref&gt;  The Regulations “protect consumers by laying down rules governing the labelling or marking of products in relation to their textile fibre content and provides uniform methods for quantitative analysis of binary textile fibre mixtures.”&lt;ref&gt;''Id''.&lt;/ref&gt;  All textiles must have a label which indicates the fibre content, in&lt;ref&gt;''Id''.&lt;/ref&gt;cluding fibre names, descriptions, and labels.&lt;ref&gt;''Id''.&lt;/ref&gt;  Any textiles sold in Ireland must comply with the Regulations.&lt;ref&gt;''Id''.&lt;/ref&gt;  The Competition and Consumer Protection Commission (“CCPC”) is the regulatory body responsible for enforcing the Regulations in Ireland.&lt;ref&gt;''Id''.&lt;/ref&gt;

&lt;u&gt;Cultural Clothing:&lt;/u&gt; There are currently no modern regulations regarding wearing Irish cultural clothing, there were historical laws which did ban certain Irish clothing.&lt;ref&gt;Marcus Harris, ''Discover the Charm of Traditional Irish Clothing | Embrace Heritage and Style'', Tartan Vibes Clothing (Jan. 5, 2024), &lt;nowiki&gt;https://www.tartanvibesclothing.com/blogs/fashion/traditional-irish-clothing?srsltid=AfmBOoouoHaNJ-0p_YXKybKf5vEOToOPhVMcioRVc5vbYK_Bx9-XZPL5&lt;/nowiki&gt;&lt;/ref&gt;  The traditional clothing in Ireland consists of a léine, typically a loose fitting linen shift that reached the knee, bróga (wool trousers), a caoat (wool jacket), and a caubeen (flat wool cap).&lt;ref&gt;''Id''.&lt;/ref&gt;  Women traditionally wore a síodóir (dress made of wool) and a brat (apron).&lt;ref&gt;''Id''.&lt;/ref&gt;  This traditional clothing can be traced back to the Irish Celts in 500BC.&lt;ref&gt;''Id''.&lt;/ref&gt;  When Christianity spread in Ireland, new traditions flooded in and in the 16th and 17th centuries, as English colonization continued, Penal Laws prohibited the Irish from wearing their traditional clothing and forced them to adopt English styles.&lt;ref&gt;''Id''.&lt;/ref&gt;  However, in the late 19th century, the Irish Cultural Movement led to an increase in interest in traditional Irish clothing and it remains an important party of the nation’s culture today, being worn on special occasions and festivals.&lt;ref&gt;''Id''.&lt;/ref&gt;

&lt;u&gt;Religious Expression and Clothing:&lt;/u&gt; Ireland does not have any laws which prohibit wearing religious clothing or symbols in public employment.&lt;ref&gt;''Religious Clothing and Symbols in Employment'', European Commission (Nov. 2017), &lt;nowiki&gt;https://www.google.com/url?sa=t&amp;source=web&amp;rct=j&amp;opi=89978449&amp;url=https://ec.europa.eu/newsroom/just/redirection/document/48810&amp;ved=2ahUKEwjc-IS-sr6RAxVPF1kFHUPeAGEQFnoECBkQAQ&amp;usg=AOvVaw3vEluFl9Q3FVWGlyg0B0QS&lt;/nowiki&gt;.&lt;/ref&gt; In 2007, there was a debate regarding the Irish police force’s uniform policy in which a Sikh man wanted to challenge a ban on wearing turbans for Reserve Police members after he was told he would not be permitted to wear his turban.&lt;ref name=&quot;:0&quot;&gt;''Id''.&lt;/ref&gt;&lt;ref name=&quot;:0&quot; /&gt;  However, the complaint was unsuccessful because the action fell outside of the conduct of the Employment Equality Acts and the issue was not addressed further.&lt;ref&gt;''Id''.&lt;/ref&gt;  In 2008, controversy arose about the use of religious symbols and clothing when a school principal asked the Minister for Education about a request by a Muslim student to wear a hijab.&lt;ref&gt;''Id''.&lt;/ref&gt;  This Irish Government chose not to issue a directive and left the decision to the schools under the Education Act 1998, but did issue a recommend that no school policy should discrimination against students of a particular religion.&lt;ref&gt;''Id''.&lt;/ref&gt;

=== Regulation of Bodily Displays and Obscenity Laws ===
&lt;u&gt;Intimate Images:&lt;/u&gt; Intimate images are defined by the Harassment, Harmful Communications and Related Offences Act 2020.&lt;ref&gt;''What is Coco’s Law? ISPCC is Offering a Free Webinar All About the Law Around the Sharing of Intimate Images'', Irish Society for the Prevention of Cruelty to Children (last visited Dec. 4, 2025), &lt;nowiki&gt;https://www.ispcc.ie/what-is-cocos-law-ispcc-is-offering-a-free-webinar-all-about-the-law-around-the-sharing-of-intimate-images/&lt;/nowiki&gt;''.''&lt;/ref&gt;  The law, also known as “Coco’s Law,” was enacted after the tragic death of Nicole “Coco” Fox, following the sharing of her intimate images online.&lt;ref&gt;''Id''.&lt;/ref&gt;  The law criminalized the making or sharing of intimate images that cause harm.&lt;ref&gt;''Sharing of Intimate Images Without Consent'', Citizens Information (last visited Dec. 4, 2025), &lt;nowiki&gt;https://www.citizensinformation.ie/en/justice/criminal-law/criminal-offences/sharing-of-intimate-images-without-consent/&lt;/nowiki&gt;.&lt;/ref&gt;  Specifically, the law makes it a crime to record, distribute, share, publish or threaten to publish intimate images without the permission of the person featured in the images and can result in a sentence up to 7 years.&lt;ref&gt;''Id''.&lt;/ref&gt;

&lt;u&gt;Nudity:&lt;/u&gt; Public indecent exposure is regulated under multiple Irish laws.&lt;ref&gt;''Law on Public Indecent Exposure'', Irish Independent (Nov. 4, 2010), &lt;nowiki&gt;https://www.independent.ie/irish-news/law-on-public-indecent-exposure/26696631.html&lt;/nowiki&gt;&lt;/ref&gt;  The Public Order Act (1994) makes it an offence in to be nude in public and can result in a fine of up to €500.&lt;ref&gt;''Id''.&lt;/ref&gt;  Section 18 of the Criminal Law (Amendment) Act (1990) also covers indecent exposure, finding an offence by “any person who commits, in public, any act in such a way as to offend modesty or cause scandal or injure the morals of the community shall be guilty . . . the person may receive a fine up to €634.87 or, if the court decides, they may be sent to prison for up to six months.”&lt;ref&gt;''Id''.&lt;/ref&gt;

With the previously heavy undertones of catholicism in Irish laws, religious attitudes led to censorship laws which also governed art.&lt;ref&gt;''The Nude'', Ask About Ireland (last visited Dec. 4, 2025), &lt;nowiki&gt;https://www.askaboutireland.ie/learning-zone/secondary-students/art/art-in-ireland/themes-in-irish-art/people-in-art/the-nude/#&lt;/nowiki&gt;.&lt;/ref&gt;  Many artists working with nude figures would work outside of Ireland when doing so to avoid the relevant laws.&lt;ref&gt;''Id''.&lt;/ref&gt;  In the 1951, Louis le Brocquy’s ''A Family'', which contained nude figures, was criticized in Ireland, but was nationally acclaimed.&lt;ref&gt;''Id''.&lt;/ref&gt;  Now, the nude appears in Irish art in many fashions and is not heavily regulated in favor of artistic expression.&lt;ref&gt;''Id''.&lt;/ref&gt; 

&lt;u&gt;Child Pornography:&lt;/u&gt; The Child Ponography Offence (2017 Act) outlaws the knowing production, distribution, transmission, dissemination, importing, exporting, selling, and supplying of child pornography.&lt;ref&gt;Criminal Law (Sexual Offences) Act 2017.&lt;/ref&gt; The Act also includes language allowing for the punishment of a person who by means of communication technology communicates with another person, including a child, for the purpose of facilitating sexual exploitation.&lt;ref&gt;''Child Pornography'', Irish Legal Blog (last visited Dec. 4, 2025), &lt;nowiki&gt;https://legalblog.ie/child-pornography/&lt;/nowiki&gt;.&lt;/ref&gt;  It also prohibits sending sexually explicit material, which is defined as “any indecent or obscene images or words.”&lt;ref&gt;''Id''.&lt;/ref&gt;

== References ==</text>
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    <title>User:Dc.samizdat/Golden chords of the 120-cell</title>
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      <comment>/* The 8-point regular polytopes */</comment>
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      <text bytes="51890" sha1="3vnsbj1mb8tqs1obny1n8rhbopsgw3c" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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 &lt;math&gt;5^2&lt;/math&gt; 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 &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Note that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length. The two disjoint squares lie in completely orthogonal central planes.]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]]. 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-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 &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; 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:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.76537,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.84776,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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 &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are indistinguishable except in the context of a rotation, where each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 90° turns.

In the 16-cell the two completely orthogonal great squares formed by the &lt;math&gt;r_2&lt;/math&gt; chords are both parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two central planes rotates that square like a wheel, while the other square does not move. The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth between the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; squares in the left and right rotational directions.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; square planes at once by the same angle, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The circular helix is a [[w:Geodesic|geodesic]] great circle on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is &lt;math&gt;4 \pi&lt;/math&gt;, and it occurs in either a left or right chiral form. We shall refer to the circular helix geodesic as an ''isocline'', and to the skew {8/3} octagram of its chords as a ''Clifford polygon''.

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The 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 square central plane to its completely orthogonal square central plane in a twisting displacement, as they tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. After 360° of rotation each vertex reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix isocline in 4-space over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle twice over the four &lt;math&gt;r_2&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each.

We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided we skewed them both in the same direction, the 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two 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.}}

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 ==
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 4 cuboctahedron central hyperplanes and 16 hexagonal central planes.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. ]]

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

The 24-cell is its own [[W:Dual polytope|dual polytope]]. Its Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}&lt;/math&gt;

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. When the 24-cell is constructed by skewing two completely orthogonal planar dodecagons, the lengths of the dodecagon chords change to: 

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

The 24-point 24-cell is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; long diameters. The &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cube long diameters.

We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix.

We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. Great hexagons are a rounder choice than great squares for the invariant rotation plane in which to rotate a 4-polytope. 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. Its Clifford polygon is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords, visible in the orthogonal projection. Two Clifford parallel skew dodecagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

In the 24-cell an isoclinic rotation by 60° in any pair of invariant completely orthogonal hexagonal central planes takes every hexagonal central plane to a Clifford parallel hexagonal central plane in a twisting displacement, as they tilt sideways 60° while rotating 60° internally. All 24 vertices move at once on two Clifford parallel circular helix 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 a circular helix isocline in 4-space over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle twice over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; 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 &amp; 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}}</text>
      <sha1>3vnsbj1mb8tqs1obny1n8rhbopsgw3c</sha1>
    </revision>
    <revision>
      <id>2810461</id>
      <parentid>2810458</parentid>
      <timestamp>2026-05-19T16:44:49Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */</comment>
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      <text bytes="52000" sha1="gym0yi5w8nq0fgu563a4h5oafazh4xa" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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 &lt;math&gt;5^2&lt;/math&gt; 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 &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Note that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length. The two disjoint squares lie in completely orthogonal central planes.]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]]. 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-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 &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; 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:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.76537,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.84776,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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 &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are indistinguishable except in the context of a rotation, where each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes &lt;math&gt;i&lt;/math&gt; 90° turns.

In the 16-cell the two completely orthogonal great squares formed by the &lt;math&gt;r_2&lt;/math&gt; chords are both parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two central planes rotates that square like a wheel, while the other square does not move. The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth between the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; squares in the left and right rotational directions.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; square planes at once by the same angle, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The circular helix is a [[w:Geodesic|geodesic]] great circle on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is &lt;math&gt;4 \pi&lt;/math&gt;, and it occurs in either a left or right chiral form. We shall refer to the circular helix geodesic as an ''isocline'', and to the skew {8/3} octagram of its chords as a ''Clifford polygon''.

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The 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 square central plane to its completely orthogonal square central plane in a twisting displacement, as they tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. After 360° of rotation each vertex reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix isocline in 4-space over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle twice over the four &lt;math&gt;r_2&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each.

We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided we skewed them both in the same direction, the 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two 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.}}

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 ==
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 4 cuboctahedron central hyperplanes and 16 hexagonal central planes.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. ]]

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

The 24-cell is its own [[W:Dual polytope|dual polytope]]. Its Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}&lt;/math&gt;

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. When the 24-cell is constructed by skewing two completely orthogonal planar dodecagons, the lengths of the dodecagon chords change to: 

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

The 24-point 24-cell is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; long diameters. The &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cube long diameters.

We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix.

We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. Great hexagons are a rounder choice than great squares for the invariant rotation plane in which to rotate a 4-polytope. 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. Its Clifford polygon is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords, visible in the orthogonal projection. Two Clifford parallel skew dodecagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

In the 24-cell an isoclinic rotation by 60° in any pair of invariant completely orthogonal hexagonal central planes takes every hexagonal central plane to a Clifford parallel hexagonal central plane in a twisting displacement, as they tilt sideways 60° while rotating 60° internally. All 24 vertices move at once on two Clifford parallel circular helix 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 a circular helix isocline in 4-space over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle twice over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; 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 &amp; 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}}</text>
      <sha1>gym0yi5w8nq0fgu563a4h5oafazh4xa</sha1>
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    <revision>
      <id>2810463</id>
      <parentid>2810461</parentid>
      <timestamp>2026-05-19T16:57:03Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */</comment>
      <origin>2810463</origin>
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      <text bytes="52000" sha1="9xxuj4em4vhd3k4yton6r66ehq7iunp" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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 &lt;math&gt;5^2&lt;/math&gt; 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 &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Note that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length. The two disjoint squares lie in completely orthogonal central planes.]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]]. 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-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 &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; 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:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.76537,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.84776,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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 &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are indistinguishable except in the context of a rotation, where each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes &lt;math&gt;i&lt;/math&gt; 90° turns.

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

In the 16-cell the two completely orthogonal great squares formed by the &lt;math&gt;r_2&lt;/math&gt; chords are both parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two central planes rotates that square like a wheel, while the other square does not move. The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth between the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; squares in the left and right rotational directions.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; square planes at once by the same angle, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The circular helix is a [[w:Geodesic|geodesic]] great circle on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is &lt;math&gt;4 \pi&lt;/math&gt;, and it occurs in either a left or right chiral form. We shall refer to the circular helix geodesic as an ''isocline'', and to the skew {8/3} octagram of its 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 square central plane to its completely orthogonal square central plane in a twisting displacement, as they tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. After 360° of rotation each vertex reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix isocline in 4-space over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle twice over the four &lt;math&gt;r_2&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each.

We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided we skewed them both in the same direction, the 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two 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.}}

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 ==
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 4 cuboctahedron central hyperplanes and 16 hexagonal central planes.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. ]]

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

The 24-cell is its own [[W:Dual polytope|dual polytope]]. Its Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}&lt;/math&gt;

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. When the 24-cell is constructed by skewing two completely orthogonal planar dodecagons, the lengths of the dodecagon chords change to: 

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

The 24-point 24-cell is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; long diameters. The &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cube long diameters.

We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix.

We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. Great hexagons are a rounder choice than great squares for the invariant rotation plane in which to rotate a 4-polytope. 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. Its Clifford polygon is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords, visible in the orthogonal projection. Two Clifford parallel skew dodecagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

In the 24-cell an isoclinic rotation by 60° in any pair of invariant completely orthogonal hexagonal central planes takes every hexagonal central plane to a Clifford parallel hexagonal central plane in a twisting displacement, as they tilt sideways 60° while rotating 60° internally. All 24 vertices move at once on two Clifford parallel circular helix 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 a circular helix isocline in 4-space over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle twice over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; 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 &amp; 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}}</text>
      <sha1>9xxuj4em4vhd3k4yton6r66ehq7iunp</sha1>
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    <revision>
      <id>2810466</id>
      <parentid>2810463</parentid>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* The 8-point regular polytopes */</comment>
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      <text bytes="51994" sha1="c4tm8s8a86e1whqmzw5mspsif7y9o90" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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 &lt;math&gt;5^2&lt;/math&gt; 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 &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Note that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length. The two disjoint squares lie in completely orthogonal central planes.]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]]. 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-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 &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; 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:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.76537,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.84776,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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 &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes &lt;math&gt;i&lt;/math&gt; 90° turns.

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

In the 16-cell the two completely orthogonal great squares formed by the &lt;math&gt;r_2&lt;/math&gt; chords are both parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two central planes rotates that square like a wheel, while the other square does not move. The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth between the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; squares in the left and right rotational directions.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; square planes at once by the same angle, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The circular helix is a [[w:Geodesic|geodesic]] great circle on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is &lt;math&gt;4 \pi&lt;/math&gt;, and it occurs in either a left or right chiral form. We shall refer to the circular helix geodesic as an ''isocline'', and to the skew {8/3} octagram of its 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 square central plane to its completely orthogonal square central plane in a twisting displacement, as they tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. After 360° of rotation each vertex reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix isocline in 4-space over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle twice over the four &lt;math&gt;r_2&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each.

We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided we skewed them both in the same direction, the 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two 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.}}

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 ==
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 4 cuboctahedron central hyperplanes and 16 hexagonal central planes.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. ]]

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

The 24-cell is its own [[W:Dual polytope|dual polytope]]. Its Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}&lt;/math&gt;

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. When the 24-cell is constructed by skewing two completely orthogonal planar dodecagons, the lengths of the dodecagon chords change to: 

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

The 24-point 24-cell is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; long diameters. The &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cube long diameters.

We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix.

We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. Great hexagons are a rounder choice than great squares for the invariant rotation plane in which to rotate a 4-polytope. 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. Its Clifford polygon is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords, visible in the orthogonal projection. Two Clifford parallel skew dodecagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

In the 24-cell an isoclinic rotation by 60° in any pair of invariant completely orthogonal hexagonal central planes takes every hexagonal central plane to a Clifford parallel hexagonal central plane in a twisting displacement, as they tilt sideways 60° while rotating 60° internally. All 24 vertices move at once on two Clifford parallel circular helix 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 a circular helix isocline in 4-space over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle twice over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; 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 &amp; 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}}</text>
      <sha1>c4tm8s8a86e1whqmzw5mspsif7y9o90</sha1>
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    <revision>
      <id>2810467</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="52036" sha1="cq3ulcqpgnd4o67aozfcilu2hqxhyzy" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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 &lt;math&gt;5^2&lt;/math&gt; 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 &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Note that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length. The two disjoint squares lie in completely orthogonal central planes.]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-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 &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; 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:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.76537,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.84776,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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 &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes &lt;math&gt;i&lt;/math&gt; 90° turns.

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

In the 16-cell the two completely orthogonal great squares formed by the &lt;math&gt;r_2&lt;/math&gt; chords are both parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two central planes rotates that square like a wheel, while the other square does not move. The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth between the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; squares in the left and right rotational directions.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; square planes at once by the same angle, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The circular helix is a [[w:Geodesic|geodesic]] great circle on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is &lt;math&gt;4 \pi&lt;/math&gt;, and it occurs in either a left or right chiral form. We shall refer to the circular helix geodesic as an ''isocline'', and to the skew {8/3} octagram of its 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 square central plane to its completely orthogonal square central plane in a twisting displacement, as they tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. After 360° of rotation each vertex reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix isocline in 4-space over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle twice over the four &lt;math&gt;r_2&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each.

We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided we skewed them both in the same direction, the 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two 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.}}

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 ==
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 4 cuboctahedron central hyperplanes and 16 hexagonal central planes.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. ]]

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

The 24-cell is its own [[W:Dual polytope|dual polytope]]. Its Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}&lt;/math&gt;

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. When the 24-cell is constructed by skewing two completely orthogonal planar dodecagons, the lengths of the dodecagon chords change to: 

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

The 24-point 24-cell is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; long diameters. The &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cube long diameters.

We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix.

We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. Great hexagons are a rounder choice than great squares for the invariant rotation plane in which to rotate a 4-polytope. 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. Its Clifford polygon is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords, visible in the orthogonal projection. Two Clifford parallel skew dodecagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

In the 24-cell an isoclinic rotation by 60° in any pair of invariant completely orthogonal hexagonal central planes takes every hexagonal central plane to a Clifford parallel hexagonal central plane in a twisting displacement, as they tilt sideways 60° while rotating 60° internally. All 24 vertices move at once on two Clifford parallel circular helix 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 a circular helix isocline in 4-space over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle twice over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; 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 &amp; 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}}</text>
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    <revision>
      <id>2810469</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="52057" sha1="hdqta4ket9f18vsva8ia8vy77kwjxrs" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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 &lt;math&gt;5^2&lt;/math&gt; 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 &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Note that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length. The two disjoint squares lie in completely orthogonal central planes.]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; 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:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.76537,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.84776,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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 &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes &lt;math&gt;i&lt;/math&gt; 90° turns.

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

In the 16-cell the two completely orthogonal great squares formed by the &lt;math&gt;r_2&lt;/math&gt; chords are both parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two central planes rotates that square like a wheel, while the other square does not move. The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth between the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; squares in the left and right rotational directions.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; square planes at once by the same angle, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The circular helix is a [[w:Geodesic|geodesic]] great circle on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is &lt;math&gt;4 \pi&lt;/math&gt;, and it occurs in either a left or right chiral form. We shall refer to the circular helix geodesic as an ''isocline'', and to the skew {8/3} octagram of its 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 square central plane to its completely orthogonal square central plane in a twisting displacement, as they tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. After 360° of rotation each vertex reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix isocline in 4-space over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle twice over the four &lt;math&gt;r_2&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each.

We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided we skewed them both in the same direction, the 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two 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.}}

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 ==
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 4 cuboctahedron central hyperplanes and 16 hexagonal central planes.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. ]]

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

The 24-cell is its own [[W:Dual polytope|dual polytope]]. Its Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}&lt;/math&gt;

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. When the 24-cell is constructed by skewing two completely orthogonal planar dodecagons, the lengths of the dodecagon chords change to: 

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

The 24-point 24-cell is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; long diameters. The &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cube long diameters.

We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix.

We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. Great hexagons are a rounder choice than great squares for the invariant rotation plane in which to rotate a 4-polytope. 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. Its Clifford polygon is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords, visible in the orthogonal projection. Two Clifford parallel skew dodecagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

In the 24-cell an isoclinic rotation by 60° in any pair of invariant completely orthogonal hexagonal central planes takes every hexagonal central plane to a Clifford parallel hexagonal central plane in a twisting displacement, as they tilt sideways 60° while rotating 60° internally. All 24 vertices move at once on two Clifford parallel circular helix 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 a circular helix isocline in 4-space over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle twice over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; 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 &amp; 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}}</text>
      <sha1>hdqta4ket9f18vsva8ia8vy77kwjxrs</sha1>
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    <revision>
      <id>2810472</id>
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      <text bytes="51936" sha1="7imvmikp4cf5int50iq2l0y2w5mslix" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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 &lt;math&gt;5^2&lt;/math&gt; 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 &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Note that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length. The two disjoint squares lie in completely orthogonal central planes.]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; 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:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.76537,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.84776,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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 &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes &lt;math&gt;i&lt;/math&gt; 90° turns.

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

In the 16-cell the two completely orthogonal great squares formed by the &lt;math&gt;r_2&lt;/math&gt; chords are both parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two central planes rotates that square like a wheel, while the other square does not move. The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth between the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; squares in the left and right rotational directions.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; square planes at once by the same angle, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The circular helix is a [[w:Geodesic|geodesic]] great circle on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is &lt;math&gt;4 \pi&lt;/math&gt;, and it occurs in either a left or right chiral form. We shall refer to the circular helix geodesic as an ''isocline'', and to the skew {8/3} octagram of its 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 square central plane to its completely orthogonal square central plane in a twisting displacement, as they tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. After 360° of rotation each vertex reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix isocline in 4-space over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle twice over the four &lt;math&gt;r_2&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each.

We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided we skewed them both in the same direction, the 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two 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.}}

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 ==
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 4 cuboctahedron central hyperplanes and 16 hexagonal central planes.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. ]]

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

The 24-cell is its own [[W:Dual polytope|dual polytope]]. Its Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}&lt;/math&gt;

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. When the 24-cell is constructed by skewing two completely orthogonal planar dodecagons, the lengths of the dodecagon chords change to: 

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

The 24-point 24-cell is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; long diameters. The &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cube long diameters.

We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix.

We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. Its Clifford polygon is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords, visible in the orthogonal projection. Two Clifford parallel skew dodecagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

In the 24-cell an isoclinic rotation by 60° in any pair of invariant completely orthogonal hexagonal central planes takes every hexagonal central plane to a Clifford parallel hexagonal central plane in a twisting displacement, as they tilt sideways 60° while rotating 60° internally. All 24 vertices move at once on two Clifford parallel circular helix 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 a circular helix isocline in 4-space over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle twice over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; 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 &amp; 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}}</text>
      <sha1>7imvmikp4cf5int50iq2l0y2w5mslix</sha1>
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    <revision>
      <id>2810475</id>
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      <text bytes="51918" sha1="k5puxuk3h1f7jeex4f34nqgplfhvf8k" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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 &lt;math&gt;5^2&lt;/math&gt; 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 &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Note that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length. The two disjoint squares lie in completely orthogonal central planes.]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; 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:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.76537,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.84776,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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 &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes &lt;math&gt;i&lt;/math&gt; 90° turns.

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

In the 16-cell the two completely orthogonal great squares formed by the &lt;math&gt;r_2&lt;/math&gt; chords are both parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two central planes rotates that square like a wheel, while the other square does not move. The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth between the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; squares in the left and right rotational directions.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; square planes at once by the same angle, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The circular helix is a [[w:Geodesic|geodesic]] great circle on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is &lt;math&gt;4 \pi&lt;/math&gt;, and it occurs in either a left or right chiral form. We shall refer to the circular helix geodesic as an ''isocline'', and to the skew {8/3} octagram of its 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 square central plane to its completely orthogonal square central plane in a twisting displacement, as they tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. After 360° of rotation each vertex reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix isocline in 4-space over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle twice over the four &lt;math&gt;r_2&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each.

We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided we skewed them both in the same direction, the 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two 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.}}

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 ==
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 4 cuboctahedron central hyperplanes and 16 hexagonal central planes.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. ]]

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

The 24-cell is its own [[W:Dual polytope|dual polytope]]. Its Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}&lt;/math&gt;

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. When the 24-cell is constructed by skewing two completely orthogonal planar dodecagons, the lengths of the dodecagon chords change to: 

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

The 24-point 24-cell is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; long diameters. The &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cube long diameters.

We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix.

We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. Its Clifford polygon is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords, visible in the orthogonal projection. Two Clifford parallel skew dodecagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

In the 24-cell an isoclinic rotation by 60° in any pair of invariant completely orthogonal hexagonal central planes takes every hexagonal central plane to a Clifford parallel hexagonal central plane in a twisting displacement, as they tilt sideways 60° while rotating 60° internally. All 24 vertices move at once on two Clifford parallel circular helix 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 a circular helix isocline in 4-space over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle twice over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

The discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; 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 &amp; 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}}</text>
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      <id>2810476</id>
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      <text bytes="51910" sha1="9yer50g7ee95zyknnut5gqxa3z7vup3" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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 &lt;math&gt;5^2&lt;/math&gt; 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 &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Note that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length. The two disjoint squares lie in completely orthogonal central planes.]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; 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:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.76537,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.84776,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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 &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes &lt;math&gt;i&lt;/math&gt; 90° turns.

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

In the 16-cell the two completely orthogonal great squares formed by the &lt;math&gt;r_2&lt;/math&gt; chords are both parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two central planes rotates that square like a wheel, while the other square does not move. The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth between the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; squares in the left and right rotational directions.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; square planes at once by the same angle, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The circular helix is a [[w:Geodesic|geodesic]] great circle on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is &lt;math&gt;4 \pi&lt;/math&gt;, and it occurs in either a left or right chiral form. We shall refer to the circular helix geodesic as an ''isocline'', and to the skew {8/3} octagram of its 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 square central plane to its completely orthogonal square central plane in a twisting displacement, as they tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. After 360° of rotation each vertex reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix isocline in 4-space over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle twice over the four &lt;math&gt;r_2&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each.

We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided we skewed them both in the same direction, the 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two 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.}}

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 ==
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 4 cuboctahedron central hyperplanes and 16 hexagonal central planes.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. ]]

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

The 24-cell is its own [[W:Dual polytope|dual polytope]]. Its Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}&lt;/math&gt;

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. When the 24-cell is constructed by skewing two completely orthogonal planar dodecagons, the lengths of the dodecagon chords change to: 

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

The 24-point 24-cell is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; long diameters. The &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cube long diameters.

We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix.

We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. Its Clifford polygon is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords, visible in the orthogonal projection. Two Clifford parallel skew dodecagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

In the 24-cell an isoclinic rotation by 60° in any pair of invariant completely orthogonal hexagonal central planes takes every hexagonal central plane to a Clifford parallel hexagonal central plane in a twisting displacement, as they tilt sideways 60° while rotating 60° internally. All 24 vertices move at once on two Clifford parallel circular helix 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 a circular helix isocline in 4-space over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle twice over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

The discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; 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 &amp; 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}}</text>
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      <text bytes="51908" sha1="gaa8pbjcxc0gqlyh37j3jas8ll6i8pk" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - May 2026}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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 &lt;math&gt;5^2&lt;/math&gt; 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 &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Note that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length. The two disjoint squares lie in completely orthogonal central planes.]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; 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:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.76537,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.84776,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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 &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes &lt;math&gt;i&lt;/math&gt; 90° turns.

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

In the 16-cell the two completely orthogonal great squares formed by the &lt;math&gt;r_2&lt;/math&gt; chords are both parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two central planes rotates that square like a wheel, while the other square does not move. The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth between the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; squares in the left and right rotational directions.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; square planes at once by the same angle, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The circular helix is a [[w:Geodesic|geodesic]] great circle on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is &lt;math&gt;4 \pi&lt;/math&gt;, and it occurs in either a left or right chiral form. We shall refer to the circular helix geodesic as an ''isocline'', and to the skew {8/3} octagram of its 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 square central plane to its completely orthogonal square central plane in a twisting displacement, as they tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. After 360° of rotation each vertex reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix isocline in 4-space over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle twice over the four &lt;math&gt;r_2&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each.

We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided we skewed them both in the same direction, the 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two 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.}}

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 ==
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 4 cuboctahedron central hyperplanes and 16 hexagonal central planes.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. ]]

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

The 24-cell is its own [[W:Dual polytope|dual polytope]]. Its Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}&lt;/math&gt;

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. When the 24-cell is constructed by skewing two completely orthogonal planar dodecagons, the lengths of the dodecagon chords change to: 

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

The 24-point 24-cell is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; long diameters. The &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cube long diameters.

We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix.

We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. Its Clifford polygon is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords, visible in the orthogonal projection. Two Clifford parallel skew dodecagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

In the 24-cell an isoclinic rotation by 60° in any pair of invariant completely orthogonal hexagonal central planes takes every hexagonal central plane to a Clifford parallel hexagonal central plane in a twisting displacement, as they tilt sideways 60° while rotating 60° internally. All 24 vertices move at once on two Clifford parallel circular helix 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 a circular helix isocline in 4-space over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle twice over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

The discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; 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 &amp; 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}}</text>
      <sha1>gaa8pbjcxc0gqlyh37j3jas8ll6i8pk</sha1>
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      <text bytes="51931" sha1="kqv4jo12prxwqp7vox0d9jftmdk9h8y" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - May 2026}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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 &lt;math&gt;5^2&lt;/math&gt; 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 &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Note that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length. The two disjoint squares lie in completely orthogonal central planes.]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; 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:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.76537,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.84776,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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 &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes &lt;math&gt;i&lt;/math&gt; 90° turns.

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

In the 16-cell the two completely orthogonal great squares formed by the &lt;math&gt;r_2&lt;/math&gt; chords are both parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two central planes rotates that square like a wheel, while the other square does not move. The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth between the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; squares in the left and right rotational directions.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; square planes at once by the same angle, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The circular helix is a [[w:Geodesic|geodesic]] great circle on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is &lt;math&gt;4 \pi&lt;/math&gt;, and it occurs in either a left or right chiral form. We shall refer to the circular helix geodesic as an ''isocline'', and to the skew {8/3} octagram of its 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 square central plane to its completely orthogonal square central plane in a twisting displacement, as they tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. After 360° of rotation each vertex reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix isocline in 4-space over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle twice over the four &lt;math&gt;r_2&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each.

We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided we skewed them both in the same direction, the 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two 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.}}

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 ==
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 4 cuboctahedron central hyperplanes and 16 hexagonal central planes.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. ]]

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

The 24-cell is its own [[W:Dual polytope|dual polytope]]. Its Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}&lt;/math&gt;

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. When the 24-cell is constructed by skewing two completely orthogonal planar dodecagons, the lengths of the dodecagon chords change to: 

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

The 24-point 24-cell is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; long diameters. The &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cube long diameters.

We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix.

We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. Its Clifford polygon is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords, visible in the orthogonal projection. Two Clifford parallel skew dodecagon geodesic orbits over &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form a circular double helix.

In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every hexagonal central plane to a Clifford parallel hexagonal central plane in a twisting displacement, as they tilt sideways 60° while rotating 60° internally. All 24 vertices move at once on two Clifford parallel circular helix 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 a circular helix isocline in 4-space over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle twice over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

The discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; 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 &amp; 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}}</text>
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    <revision>
      <id>2810492</id>
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      <text bytes="51736" sha1="lnbloujvzbebndgzyj71iklccv8hq84" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - May 2026}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;math&gt;\{5,3,3\}&lt;/math&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;.{{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 &lt;math&gt;5^2&lt;/math&gt; 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 &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Note that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length. The two disjoint squares lie in completely orthogonal central planes.]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.76537,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.84776,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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 &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes &lt;math&gt;i&lt;/math&gt; 90° turns.

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

In the 16-cell the two completely orthogonal great squares formed by the &lt;math&gt;r_2&lt;/math&gt; chords are both parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two central planes rotates that square like a wheel, while the other square does not move. The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth between the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; squares in the left and right rotational directions.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; square planes at once by the same angle, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The circular helix is a [[w:Geodesic|geodesic]] great circle on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is &lt;math&gt;4 \pi&lt;/math&gt;, and it occurs in either a left or right chiral form. We shall refer to the circular helix geodesic as an ''isocline'', and to the skew {8/3} octagram of its 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 square central plane to its completely orthogonal square central plane in a twisting displacement, as they tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. After 360° of rotation each vertex reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix isocline in 4-space over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle twice over the four &lt;math&gt;r_2&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each.

We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided we skewed them both in the same direction, the 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two 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.}}

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 ==
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 4 cuboctahedron central hyperplanes and 16 hexagonal central planes.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. ]]

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

The 24-cell is its own [[W:Dual polytope|dual polytope]]. Its Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}&lt;/math&gt;

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. When the 24-cell is constructed by skewing two completely orthogonal planar dodecagons, the lengths of the dodecagon chords change to: 

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

The 24-point 24-cell is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cube long diameters.

We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix.

We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. Its Clifford polygon is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords, visible in the orthogonal projection. Two Clifford parallel skew dodecagon geodesic orbits over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix.

In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every hexagonal central plane to a Clifford parallel hexagonal central plane in a twisting displacement, as they tilt sideways 60° while rotating 60° internally. All 24 vertices move at once on two Clifford parallel circular helix 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 a circular helix isocline in 4-space over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle twice over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

The discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; 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 &amp; 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}}</text>
      <sha1>lnbloujvzbebndgzyj71iklccv8hq84</sha1>
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    <revision>
      <id>2810493</id>
      <parentid>2810492</parentid>
      <timestamp>2026-05-19T19:13:37Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Compounds in the 120-cell */</comment>
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      <text bytes="51718" sha1="i8u1hkjwxt3z8ri7db2qpe6jxjp1dvd" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - May 2026}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;math&gt;\{5,3,3\}&lt;/math&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Note that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length. The two disjoint squares lie in completely orthogonal central planes.]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.76537,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.84776,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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 &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes &lt;math&gt;i&lt;/math&gt; 90° turns.

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

In the 16-cell the two completely orthogonal great squares formed by the &lt;math&gt;r_2&lt;/math&gt; chords are both parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two central planes rotates that square like a wheel, while the other square does not move. The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth between the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; squares in the left and right rotational directions.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; square planes at once by the same angle, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The circular helix is a [[w:Geodesic|geodesic]] great circle on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is &lt;math&gt;4 \pi&lt;/math&gt;, and it occurs in either a left or right chiral form. We shall refer to the circular helix geodesic as an ''isocline'', and to the skew {8/3} octagram of its 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 square central plane to its completely orthogonal square central plane in a twisting displacement, as they tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. After 360° of rotation each vertex reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix isocline in 4-space over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle twice over the four &lt;math&gt;r_2&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each.

We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided we skewed them both in the same direction, the 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two 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.}}

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 ==
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 4 cuboctahedron central hyperplanes and 16 hexagonal central planes.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. ]]

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

The 24-cell is its own [[W:Dual polytope|dual polytope]]. Its Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}&lt;/math&gt;

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. When the 24-cell is constructed by skewing two completely orthogonal planar dodecagons, the lengths of the dodecagon chords change to: 

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

The 24-point 24-cell is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cube long diameters.

We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix.

We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. Its Clifford polygon is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords, visible in the orthogonal projection. Two Clifford parallel skew dodecagon geodesic orbits over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix.

In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every hexagonal central plane to a Clifford parallel hexagonal central plane in a twisting displacement, as they tilt sideways 60° while rotating 60° internally. All 24 vertices move at once on two Clifford parallel circular helix 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 a circular helix isocline in 4-space over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle twice over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

The discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; 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 &amp; 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}}</text>
      <sha1>i8u1hkjwxt3z8ri7db2qpe6jxjp1dvd</sha1>
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    <revision>
      <id>2810495</id>
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      <text bytes="51753" sha1="bsiqd6t13emg6rmp9mq4u6lxre3hlpo" xml:space="preserve">= Golden chords of the 120-cell =

{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - May 2026}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;math&gt;\{5,3,3\}&lt;/math&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
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|-
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|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Note that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length. The two disjoint squares lie in completely orthogonal central planes.]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.76537,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.84776,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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 &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes &lt;math&gt;i&lt;/math&gt; 90° turns.

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

In the 16-cell the two completely orthogonal great squares formed by the &lt;math&gt;r_2&lt;/math&gt; chords are both parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two central planes rotates that square like a wheel, while the other square does not move. The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth between the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; squares in the left and right rotational directions.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal &lt;math&gt;r_2&lt;/math&gt; square planes at once by the same angle, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The circular helix is a [[w:Geodesic|geodesic]] great circle on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is &lt;math&gt;4 \pi&lt;/math&gt;, and it occurs in either a left or right chiral form. We shall refer to the circular helix geodesic as an ''isocline'', and to the skew {8/3} octagram of its 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 square central plane to its completely orthogonal square central plane in a twisting displacement, as they tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. After 360° of rotation each vertex reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix isocline in 4-space over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle twice over the four &lt;math&gt;r_2&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each.

We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided we skewed them both in the same direction, the 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two 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.}}

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 ==
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 4 cuboctahedron central hyperplanes and 16 hexagonal central planes.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. ]]

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

The 24-cell is its own [[W:Dual polytope|dual polytope]]. Its Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}&lt;/math&gt;

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. When the 24-cell is constructed by skewing two completely orthogonal planar dodecagons, the lengths of the dodecagon chords change to: 

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

The 24-point 24-cell is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cube long diameters.

We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix.

We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. Its Clifford polygon is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords, visible in the orthogonal projection. Two Clifford parallel skew dodecagon geodesic orbits over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix.

In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every hexagonal central plane to a Clifford parallel hexagonal central plane in a twisting displacement, as they tilt sideways 60° while rotating 60° internally. All 24 vertices move at once on two Clifford parallel circular helix 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 a circular helix isocline in 4-space over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle twice over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

The discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; 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 &amp; 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}}</text>
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      <text bytes="4396" sha1="m0p2c1h59hg5i8kdhy34sti6oftft9q" xml:space="preserve">Welcome to Introduction to Modern Hebrew, the first course in a 7-part series designed to bring the beginner Hebrew speaker to fluency. This course will cover the Hebrew alphabet and reading Hebrew, basic grammar, and short introductions and questions.

==Hebrew alphabet==
Unlike the English or Latin alphabet, the Hebrew alphabet is written from right to left. Also, all of the letters in the Hebrew alphabet are consonants; the vowels, or ניקוד (nikkud) are small symbols above or below the letters. The consonants are shown below:

{| class=&quot;wikitable&quot;
|+ Hebrew alphabet
|-
! Letter !! Final form (if exists) !! Sound !!Name
|-
| א || || No sound/Silent||Aleph
|-
| בּ ||  || B as in bee||Bet
|-
|ב|| || V as in Violin||Vet
|-
| ג ||  || G as in goat||Gimel
|-
| ד ||  || D as in dog||Dalet
|-
| ה ||  || H as in Hat, or silent if at the end of a word||Hei
|-
| ו ||  || V as in violin, or O/U depending on the vowel applied||Vav
|-
| ז ||  || Z as in Zebra||Zayin
|-
| ח ||  || KH as in Loch or Nacht||Khet
|-
| ט ||  || T as in Tart||Tet
|-
| י ||  || I or Y||Yud
|-
|כּ||||K as in Kitten||Kaf
|-
| כ || ך || KH as in Loch or Nacht||Khaf
|-
| ל ||  || L as in Lion||Lamed
|-
| מ || ם || M as in Music||Mem
|-
| נ || ן || N as in Nordic||Nun
|-
| ס ||  || S as in Sail||Samech
|-
| ע ||  || Glottal stop (')||Ayin
|-
| פּ||||P||Pei
|-
| פ || ף || F|| Fei
|-
| צ || ץ || TS as in Arts|| Tzadik
|-
| ק ||  || K as in King (sometimes transcribed Q)||Quf
|-
| ר ||  || R as in Rain||Resh
|-
| שׁ ||  || Sh as in Shard||Shin
|-
| שׂ|| || S as in Song||Sin
|-
| ת ||  || T as in Tart||Taf
|}
If the letters כ, מ, נ, פ, or צ come at the end of a word, they are replaced with their final form counterparts.
===Vowels===
The vowels in Hebrew language are small symbols shown on the Hebrew letter called nikud. The sound of the letter will depend on the nikud that the letter has, for example:
*אַ, pronounced ''A'' as in Father
*אֶ or אֵ, pronounced ''E'' as in Bed
*אִ or usually אִי, pronounced ''I'' as in Machine
*אֻ or usually אוּ, pronounced ''U'' as in Tube
*אֹ or usually אוֹ, pronoucned ''O'' as in Rope

You can check how different letters sound with different nikud in the free [https://hebrewmastery.com/tools/pronunciation-audio Hebrew Pronunciation Tool].
'''A note on the appearance of vowels:''' nikud in modern Hebrew is usually used in works for beginners; in content such as newspapers, textbooks, and most books and novels the vowels are omitted, and a word is recognized based off of the consonants and the context of the whole sentence.

===Quiz===
&lt;quiz display=&quot;simple&quot;&gt;
{How is בַנַנַה pronounced?}
- Gararim
- Atatah
+ Bananah
- Ananas


{What is the Hebrew spelling of the word &quot;Ananas&quot;?}
+ אַנַנַס
- בַנַנַה
-אַלַנֶס 
– אַלַנֶס
- נוֹשַה

{What sound does the Hebrew letter בּ represent?}
-T
-R
+B
-P

{What sound does the Hebrew vowel אֶ represent?}
-A
-U
+E
-O

{Which letter is silent?}
-Bet
+Aleph
-Resh
-Tet

{How is the word שָׁלוֹם read?}
-Shulam
-Shilem
-Salom
+Shalom

{How is the letter ר pronounced?}
+R as in Race
-D as in Dog
-M as in Master
-N as in Not
&lt;/quiz&gt;
==Israeli pronunciation==
The Israeli pronunciation of a few consonants is different from their English counterparts:
*To pronounce ל, Lamed, put your tongue at the roof of your mouth and then say a normal &quot;l&quot; sound.
*ר, Resh, is pronounced a few different ways in Israel. The main pronunciation is pronouncing the R from the throat like in French, but in many cases speakers will roll the R like in  Spanish or Italian.
==Basic greetings and introductions==
{| class=&quot;wikitable sortable&quot;
|+Basic greetings and introductions
!Hebrew
!Hebrew transliteration
!English
!Notes
|-
|שׁלום
|shalom
|Hello/Goodbye
|Shalom is a versatile word in Hebrew, having three different meanings that are all common in everyday use. It can mean Hello, Goodbye, or Peace.
|-
|בוקר טוב
|boker tov
|Good morning
|Tov means good, and Boker means morning.
|-
|ערב טוב
|erev tov
|Good evening
|
|-
|להתראות
|lehitraot
|Goodbye
|More common than Shalom, but both are still used.
|-
|מה שלומך
|ma shlomkha?
|How are you?
|
|-
|אני
|ani
|I/I am
|
|-
|איך קוראים לך
|Ekh kor'im lekha?
|What is your name?
|lit. &quot;How are you called?&quot;
|}
[[Category:Hebrew]]</text>
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    <title>Law School 101/Legal Reasoning &amp; The Survey Course/Class 2 - Introduction to Torts: Negligence &amp; The &quot;Freak Accident&quot;</title>
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      <text bytes="5893" sha1="2qvh7llozf9jvma5zk1g7tyccgts3xe" xml:space="preserve">[[File:Artists-impressions-of-Lady-Justice, (statue on the Old Bailey, London).png|center|frameless|909x909px]]

== Class 2 - Introduction to Torts: Negligence &amp; The &quot;Freak Accident&quot; ==

=== Objective ===
In Class 1, we learned about Property (who owns what). Today, we look at Torts (what happens when someone gets hurt). You will learn the &quot;Periodic Table&quot; of injury law: The 4 Elements of Negligence.

== PART I: The Lecture ==

=== What is a Tort? ===
A &quot;Tort&quot; (French for &quot;wrong&quot;) is a civil claim where one person causes harm to another. It is not Criminal Law.

* '''Criminal Law:''' The State vs. You. (Goal: Punishment/Jail).
* '''Tort Law:''' Me vs. You. (Goal: Money/Compensation).

Most torts are accidents. We call these '''Negligence'''.

=== The Formula for Negligence ===
In Class 1, we learned IRAC. In Torts, the &quot;R&quot; (Rule) is almost always the same four elements. To win a lawsuit for an accident, the Plaintiff must prove '''all four''' of these:

# '''Duty:''' Did the Defendant owe a legal obligation to keep the Plaintiff safe?
# '''Breach:''' Did the Defendant fail to act like a &quot;Reasonable Person&quot;?
# '''Causation:''' Did the Defendant's failure actually cause the injury?
# '''Damages:''' Was the Plaintiff actually hurt?

Today, we focus on '''Element #1: Duty'''. You do not have a duty to keep the whole world safe. You generally only have a duty to avoid '''foreseeable''' harm.

== PART II: The Case Study ==
This is arguably the most famous case in American legal history. It is the &quot;Jedi Master&quot; level of legal reasoning regarding causation and duty.

'''The Case:''' ''Palsgraf v. Long Island Railroad Co.'' (New York Court of Appeals, 1928)

'''The Facts (The Rube Goldberg Machine):''' Picture a train station in 1924.

# A man is running to catch a moving train.
# Two railroad guards try to help him aboard. One pulls him from the train, the other pushes him from behind.
# In the scuffle, the man drops a package wrapped in nondescript newspaper.
# '''The Twist:''' The package contained fireworks.
# The package hits the rails and explodes.
# The shockwave from the explosion knocks over heavy coin-operated scales at the ''other end'' of the platform.
# The scales fall on '''Mrs. Palsgraf''', injuring her.

Mrs. Palsgraf sues the Railroad. She argues: ''Your guards were negligent in pushing that man. Because they were negligent, the explosion happened, and I got hurt.''

'''The Legal Issue:''' The guards were definitely negligent toward the man with the package (you shouldn't push people onto trains). But did they owe a '''Duty of Care''' to Mrs. Palsgraf, who was standing far away and was injured by a freak chain of events?

== PART III: Your Turn (The Cold Call) ==
The core question is about the '''Scope of Liability'''. How far does the blame ripple out?

'''Argument A (For Mrs. Palsgraf - The &quot;But For&quot; Argument):''' If the guards hadn't pushed the man, the package wouldn't have fallen. If the package hadn't fallen, Mrs. Palsgraf wouldn't have been hit. The guards started the chain reaction. If you do something stupid (breach), shouldn't you pay for ''all'' the damage that results?

'''Argument B (For the Railroad - The &quot;Foreseeability&quot; Argument):''' The guards saw a newspaper-wrapped package. They had no way of knowing it was a bomb. Pushing a man might hurt ''him'', but no &quot;Reasonable Person&quot; would guess that pushing a man would cause scales to fall on a lady 30 feet away. How can you have a duty to prevent something you can't predict?

'''Make your decision.''' Does the Railroad owe Mrs. Palsgraf money?

.

.

.

== PART IV: The Verdict ==
'''The Court ruled in favor of the Railroad (Defendant). Mrs. Palsgraf gets nothing.'''

This opinion was written by '''Justice Benjamin Cardozo''', a legend of American law. He established the '''&quot;Zone of Danger&quot;''' test.

* '''The Ruling:''' Negligence is not just &quot;doing something wrong.&quot; It is doing something wrong ''in relation to a specific person''.
* '''The Analysis:'''
** If the guards had knocked over the man and ''he'' broke his leg, the Railroad would be liable. That is a '''foreseeable''' risk of pushing someone.
** However, there was nothing in the situation to suggest that the package was dangerous to persons far away.
** Cardozo wrote: ''&quot;The risk reasonably to be perceived defines the duty to be obeyed.&quot;''
** Because Mrs. Palsgraf was standing outside the '''&quot;Zone of Danger&quot;''' (the area where a reasonable person would expect injury to occur from that specific action), the Railroad owed her no duty.
'''The Dissent (Justice Andrews):''' Justice Andrews disagreed completely. He argued for a broader view: '''Proximate Cause'''. He believed that if you commit a wrongful act, you are responsible for the natural consequences of that act, even if they are weird. He famously wrote regarding where to draw the line: ''&quot;It is all a question of expediency... a rough sense of justice.&quot;''

== PART V: Homework / Takeaway ==
The concept of '''Foreseeability''' is the shield that protects companies and people from infinite lawsuits.

'''Your &quot;Homework&quot; Thought Experiment:''' You are driving your car and you text on your phone (Breach of Duty). You swerve and hit a telephone pole. The pole falls. It causes a power outage in the neighborhood. Two blocks away, a surgeon is performing an operation. The lights go out. The backup generator fails. The patient suffers an injury.

The patient sues '''you''' (the texting driver).

Apply the ''Palsgraf'' rule:

# Was the patient in the &quot;Zone of Danger&quot; of your car?
# Was it '''foreseeable''' that texting would cause a surgical error two blocks away?

''(Law School Answer: Most courts would say No. You are liable for the pole and the car, but the surgical injury is likely too remote/unforeseeable to create a Duty.)''
[[File:Horizontal bar decorative.jpg|center|frameless]]

[[Category:Law School 101]]</text>
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    <title>User:Atcovi/OGM &amp; Suicide/The Paper</title>
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      <text bytes="17657" sha1="9nbvcaqx0rr262o3ro2g1no4cwkvyhr" xml:space="preserve">'''Major objective''': OGM may function as a context-dependent cognitive vulnerability that contributes to suicidal escalation through mechanisms embedded within the IMV model, particularly in high-risk populations. Through mechanisms outlined in the IMV framework, OGM may contribute to increased suicidal ideation within high-risked populations. 

==Introduction==
'''[[w:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory]]''' (OGM) describes a reduced ability to recall specific events in one's autobiographical memory. For example, one may remember attending a birthday party at some point in their life, but they could not uniquely recall a specific instance of attending a birthday party. OGM has been empirically associated with depression, with depressed individuals reporting higher levels of OGM than non-depressed individuals&lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Sumner|first=Jennifer A.|last2=Griffith|first2=James W.|last3=Mineka|first3=Susan|date=2010-07|title=Overgeneral autobiographical memory as a predictor of the course of depression: a meta-analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC2878838/|journal=Behaviour Research and Therapy|volume=48|issue=7|pages=614–625|doi=10.1016/j.brat.2010.03.013|issn=1873-622X|pmc=2878838|pmid=20399418}}&lt;/ref&gt;. Given the association of depression and suicidal ideation&lt;ref&gt;{{Cite journal|last=Chachamovich|first=Eduardo|last2=Stefanello|first2=Sabrina|last3=Botega|first3=Neury|last4=Turecki|first4=Gustavo|date=2009-05|title=[Which are the recent clinical findings regarding the association between depression and suicide?]|url=https://pubmed.ncbi.nlm.nih.gov/19565147|journal=Revista Brasileira De Psiquiatria (Sao Paulo, Brazil: 1999)|volume=31 Suppl 1|pages=S18–25|doi=10.1590/s1516-44462009000500004|issn=1516-4446|pmid=19565147}}&lt;/ref&gt;, utilizing the '''Integrated Motivational-Volitional (IMV) model''' provides a theoretical cognitive framework to argue that OGM intensifies suicidal escalation through moderators described in the model. 

The IMV model portrays suicidal behavior as an escalating, behavioral process divided into three phases: pre-motivational phase, motivational phase, and volitional phase. The motivational phase is characterized by suicidal ideation formation, where feelings of entrapment (described as a &quot;proximal [predictor] of suicidal ideation&quot;&lt;ref&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational-volitional model of suicidal behaviour|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/|journal=Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=1471-2970|pmc=6053985|pmid=30012735}}&lt;/ref&gt;), poor problem-solving abilities, brooding, and interpersonal vulnerabilities (thwarted belongingness and perceived burdensomeness) may transition the individual to the volitional phase. When looking at the IMV model and assessing where OGM could play a part in the transition to suicidal ideation, OGM may impair problem-solving capabilities and the ability to learn from the past through reduced retrieval of specific past experiences, leading to hopelessness&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Jiang|first=Wen|last2=Hu|first2=Guangtao|last3=Zhang|first3=Jingxuan|last4=Chen|first4=Ken|last5=Fan|first5=Dongni|last6=Feng|first6=Zhengzhi|date=2020-10-12|title=Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people|url=https://doi.org/10.1186/s12888-020-02877-6|journal=BMC Psychiatry|language=en|volume=20|issue=1|pages=501|doi=10.1186/s12888-020-02877-6|issn=1471-244X|pmc=7549224|pmid=33046032}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5}}&lt;/ref&gt;.

The current literature aims to shed light on a neglected niche of suicide research: autobiographical memory. Despite the overwhelming research suggesting correlations between OGM and depression and suicidal ideation, research has not thoroughly explored OGM's exact role in a cognitive, theoretical framework of suicidal ideation (specifically within the IMV model). By conducting a narrative review and integrating research on OGM's role in suicidal ideation, this paper furthers understanding on OGM's role in suicidal ideation within the IMV framework. 

==OGM as a Vulnerability==
Evidence suggests that OGM is a cognitive vulnerability associated with depression and suicidal ideation, though its predictive relevance may vary depending on the population. 

A meta analysis performed by Sumner et. al (2010) found that OGM accounted for about 1-2% of the variance in depressive symptoms at follow-up&lt;ref name=&quot;:2&quot; /&gt;. A 2020 study found that OGM was associated with depressed patients' current suicidal ideation state and worse-point suicidal ideation, while OGM affected the healthy patients' worse-point suicidal ideation&lt;ref name=&quot;:2&quot; /&gt;. However, Crane et. al (2016) conducted a longitudinal study of n≈5800 adolescents from ages 13 to 16 and found that OGM was not significantly associated with depression and did not moderate the effect of life events, suggesting OGM may not be as generalizable to community samples vs. high-risk populations&lt;ref name=&quot;:2&quot; /&gt;. Even though the findings indicated that OGM was not significantly associated with depression, the study highlights that OGM may function as a vulnerability within high-risk populations. This suggests that OGM's predictive relevance may depend on the risk-level of the population.

OGM persists even past depression, as found in Hallford et. al (2022). A meta-analysis indicated that participants with remitted depression continue to experience small to moderate deficiencies in being able to recall &quot;specific, event-level personal memories&quot;. This indicates that OGM isn't merely a symptom of depression, but may function as a risk factor for future depressive episodes&lt;ref name=&quot;:2&quot; /&gt;.    

Considering the above, the analysis suggests that OGM may function as a cognitive vulnerability that is further exacerbated in high-risk populations vs. the general population.    

==Mechanisms==
The inability to retrieve specific memories from one's autobiographical memory may lead to inefficient problem-solving abilities, which can impact one's ability to deal with difficult situations as they lack past experiences to rely on&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Arie|first=Miri|last2=Apter|first2=Alan|last3=Orbach|first3=Israel|last4=Yefet|first4=Yael|last5=Zalzman|first5=Gil|date=2008-01-01|title=Autobiographical memory, interpersonal problem solving, and suicidal behavior in adolescent inpatients|url=https://www.sciencedirect.com/science/article/pii/S0010440X07000922|journal=Comprehensive Psychiatry|volume=49|issue=1|pages=22–29|doi=10.1016/j.comppsych.2007.07.004|issn=0010-440X}}&lt;/ref&gt;. Failing to deal with difficult situations can drive an individual to hopelessness. In 1986, an article by Mark J. Williams and Keith Broadbent stipulated that individuals who recently attempted suicide had biased latencies in autobiographical memory retrieval and had reduced specificity in responses to especially positive cues&lt;ref&gt;{{Cite web|url=https://www.tandfonline.com/action/cookieAbsent|website=www.tandfonline.com|doi=10.1080/02699938808410925|access-date=2026-05-08}}&lt;/ref&gt;. OGM may not have a direct effect on suicide, but the evidence suggests that OGM intensifies the risk of suicidal ideation through deteriorating cognitive functioning.   

Rumination involves maladaptive dwelling on one's past negative emotions and feelings&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Treynor|first=Wendy|last2=Gonzalez|first2=Richard|last3=Nolen-Hoeksema|first3=Susan|date=2003-06-01|title=Rumination Reconsidered: A Psychometric Analysis|url=https://doi.org/10.1023/A:1023910315561|journal=Cognitive Therapy and Research|language=en|volume=27|issue=3|pages=247–259|doi=10.1023/A:1023910315561|issn=1573-2819}}&lt;/ref&gt;, and is associated with suicidal behavior/ideation&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational–volitional model of suicidal behaviour|url=https://royalsocietypublishing.org/doi/10.1098/rstb.2017.0268|journal=Philosophical Transactions of the Royal Society B: Biological Sciences|language=en|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=0962-8436|pmc=6053985|pmid=30012735}}&lt;/ref&gt;. One may reflect on their negative emotions and question such emotions in an abstract manner (&quot;How did I get to feel this way?&quot;&lt;ref name=&quot;:0&quot; /&gt;, &quot;Why did this happen to me?&quot;&lt;ref&gt;{{Cite journal|last=Sumner|first=Jennifer A.|date=2012-02|title=The mechanisms underlying overgeneral autobiographical memory: an evaluative review of evidence for the CaR-FA-X model|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3246105/|journal=Clinical Psychology Review|volume=32|issue=1|pages=34–48|doi=10.1016/j.cpr.2011.10.003|issn=1873-7811|pmc=3246105|pmid=22142837}}&lt;/ref&gt;), which causes one's memory retrieval to capture negative intermediate conceptual information (ex, &quot;I'm a failure&quot;) instead of specific memories. Repeated rumination strengthens these negative self-beliefs, leading to frequent capture of negative conceptual themes and impeding memory retrieval. This association of rumination and general memory aligns with the CaR-FA-X model&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;.  

Within the IMV model, we propose that OGM falls under '''Threat to Self Moderators (TSM)'''. TSMs spur entrapment (a perceived sense of being trapped by defeat/humilitation), which can lead to suicidal ideation depending on the effects of '''Motivational Moderators (MM).''' Positive factors, such as motivation to live, positive future thinking, and belongingess can offset the transition of entrapment → suicidal ideation, though negative factors, such as thwarted belongingness, very little social support, and perceived burdensomeness, may increase the chance of entrapment converting into suicidal ideation&lt;ref name=&quot;:1&quot; /&gt;.  

==OGM → Suicidal ideation (CORE)==
The research suggests that OGM is not just associated with depression, but is a contributing factor towards suicidal ideation (especially in high-risk populations). Jiang et. al (2020) found in a study of 365 participants, with roughly 51% of the participants clinically depressed while the other roughly 49% of participants were classified as &quot;healthy&quot;, that OGM had an increased presence in the depressed group vs. the &quot;healthy&quot; group. WSI (worst suicidal thoughts one has ever had [at a certain point]) and CSI (current point of suicidal ideation) were significantly affected by OGM in the depressed group. OGM was also found to be a mediator between CSI and childhood trauma in depressed patients. As OGM leads to negative memory biases and, therefore, the maintenance of a negative mental state, the researchers suggested that OGM may be a consistent contributor to suicidal ideation in depressed patients&lt;ref name=&quot;:3&quot; /&gt;. These findings are corroborated by a more recent (2025) study on depressed patients with varying levels of SI, where the researchers concluded that OGM may be &quot;a maladaptive cognitive avoidance strategy&quot; rather than simply a deterioration in memory. Zhu et. al (2025) further explain that individuals with OGM have a difficult time recalling positive memories, which reinforce negative recollections, spurring hopelessness and the transition to suicidal ideation&lt;ref&gt;{{Cite journal|last=Zhu|first=Ying|last2=Yin|first2=Qianlan|last3=Xu|first3=Huijing|last4=Xiao|first4=Fang|last5=Jiang|first5=Qian|last6=Liang|first6=Meng|last7=Cheng|first7=Qi|last8=Liu|first8=Taosheng|date=2025-11-24|title=Speech feature identification model for depressed individuals with suicidal ideation based on autobiographical memory|url=https://doi.org/10.1186/s12888-025-07635-0|journal=BMC Psychiatry|language=en|volume=25|issue=1|pages=1154|doi=10.1186/s12888-025-07635-0|issn=1471-244X|pmc=12713288|pmid=41286790}}&lt;/ref&gt;.

In a 2008 study on autobiographical memory, interpersonal problem-solving skills, and suicidal behaviour in adolescents and young adults, Arie et. al (2008) found that OGM was significantly associated with hopelessness and poor problem-solving abilities in adolescents. This suggests that being able to retrieve specific memories in one's autobiographical memory improves problem-solving skills, as they are able to draw back from past experiences to address challenging interpersonal situations&lt;ref name=&quot;:4&quot; /&gt;. Kaviani et. al (2011) found that depressed individuals with more severe suicidal ideation levels had more difficulty in retrieving specific thoughts in comparison to depressed individuals with less severe suicidal ideation&lt;ref&gt;{{Cite journal|last=Kaviani|first=H.|last2=Rahimi|first2=M.|last3=Rahimi-Darabad|first3=P.|last4=Naghavi|first4=K. Kamyar H.|date=2003|title=HOW AUTOBIOGRAPHICAL MEMORY DEFICITS AFFECT PROBLEM-SOLVING IN DEPRESSED PATIENTS|url=https://acta.tums.ac.ir/index.php/acta/article/view/2663|journal=Acta Medica Iranica|language=en-US|pages=194–198|issn=1735-9694}}&lt;/ref&gt;. These findings suggest that OGM plays a unique factor in contributing to suicidal ideation through maladaptive cognitive processing rather than being merely a symptom of depression.

==Contradictions / Nuances==
Despite the importance of OGM and the findings indicating its potential amplification of suicidal ideation, OGM does not appear to be a consistent detriment in low-risk populations. 

Crane et. al (2016) conducted a longitudinal study of 5792 adolescents from ages 13 to 16 and found no significant findings that OGM played a direct or interactive role with depression, suicidal ideation, and self-harm (when accounted for confounding variables) in a general population&lt;ref&gt;{{Cite journal|last=Crane|first=Catherine|last2=Heron|first2=Jon|last3=Gunnell|first3=David|last4=Lewis|first4=Glyn|last5=Evans|first5=Jonathan|last6=Williams|first6=J. Mark G.|date=2016|title=Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort study|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/|journal=Memory (Hove, England)|volume=24|issue=3|pages=348–363|doi=10.1080/09658211.2015.1008014|issn=1464-0686|pmc=4743605|pmid=25716137}}&lt;/ref&gt;. The authors concluded that OGM appears to be more clinically meaningful in high-risk populations that are already cognitively vulnerable through depression and/or psychopathology. The OGM x Stress interaction theory is supported by another longitudinal study done on 174 Caucasian adolescents by Stange et. al (2012), where they found that OGM was found to be a vulnerability to adolescents with depression (especially emotional maltreatment)&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;. 

These findings, alongside with Jiang et. al (2020), suggest that OGM is context-dependent and may play a significant role in the development of depression and/or suicidal ideation if the individual is already susceptible for depression and/or mental disorders. This aligns with the diathesis-stress model, suggesting a set of factors interact with pre-existing vulnerabilities to produce a &quot;disordered state&quot;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5|pages=24-25}}&lt;/ref&gt;.

==Conclusion==
OGM has been found to be a cognitive vulnerability for suicidal ideation amongst clinically high-risk samples. Within the IMV model, OGM may be a TSM (Threat to Self Moderators), which may contribute to entrapment, hopelessness, negative Motivational Moderators (such as thwarted belongingness and perceived burdensomeness), and the transition from suicidal ideation to suicidal behavior within the Volitional Phase (where suicidal behavior is present). Despite significant research indicating OGM's unique contribution to suicidal ideation, OGM may be a significant contributor in high-risk/depressed populations. Future research should look into OGM's place in suicidal models and assess whether therapies targeting OGM, such as Memory Specificity Training (MEST), would be beneficial across diverse populations in reducing the risk of suicidal ideation. 

== References ==
{{Reflist}}
[[Category:Atcovi/OGM &amp; Suicide Poster]]</text>
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      <text bytes="17622" sha1="dq8ql3q0guwjt7vqfshxqi63c2fvx5v" xml:space="preserve">'''Major objective''': OGM may function as a context-dependent cognitive vulnerability that contributes to suicidal escalation through mechanisms embedded within the IMV model, particularly in high-risk populations. Through mechanisms outlined in the IMV framework, OGM may contribute to increased suicidal ideation within high-risked populations. 

==Introduction==
'''[[w:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory]]''' (OGM) describes a reduced ability to recall specific events in one's autobiographical memory. For example, one may remember attending a birthday party at some point in their life, but they could not uniquely recall a specific instance of attending a birthday party. OGM has been empirically associated with depression, with depressed individuals reporting higher levels of OGM than non-depressed individuals&lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Sumner|first=Jennifer A.|last2=Griffith|first2=James W.|last3=Mineka|first3=Susan|date=2010-07|title=Overgeneral autobiographical memory as a predictor of the course of depression: a meta-analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC2878838/|journal=Behaviour Research and Therapy|volume=48|issue=7|pages=614–625|doi=10.1016/j.brat.2010.03.013|issn=1873-622X|pmc=2878838|pmid=20399418}}&lt;/ref&gt;. Given the association of depression and suicidal ideation&lt;ref&gt;{{Cite journal|last=Chachamovich|first=Eduardo|last2=Stefanello|first2=Sabrina|last3=Botega|first3=Neury|last4=Turecki|first4=Gustavo|date=2009-05|title=[Which are the recent clinical findings regarding the association between depression and suicide?]|url=https://pubmed.ncbi.nlm.nih.gov/19565147|journal=Revista Brasileira De Psiquiatria (Sao Paulo, Brazil: 1999)|volume=31 Suppl 1|pages=S18–25|doi=10.1590/s1516-44462009000500004|issn=1516-4446|pmid=19565147}}&lt;/ref&gt;, utilizing the '''Integrated Motivational-Volitional (IMV) model''' provides a theoretical cognitive framework to argue that OGM intensifies suicidal escalation through moderators described in the model. 

The IMV model portrays suicidal behavior as an escalating, behavioral process divided into three phases: pre-motivational phase, motivational phase, and volitional phase. The motivational phase is characterized by suicidal ideation formation, where feelings of entrapment (described as a &quot;proximal [predictor] of suicidal ideation&quot;&lt;ref&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational-volitional model of suicidal behaviour|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/|journal=Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=1471-2970|pmc=6053985|pmid=30012735}}&lt;/ref&gt;), poor problem-solving abilities, brooding, and interpersonal vulnerabilities (thwarted belongingness and perceived burdensomeness) may transition the individual to the volitional phase. When looking at the IMV model and assessing where OGM contributes to suicidal ideation, OGM appears to impair problem-solving capabilities and the ability to learn from the past through reduced retrieval of specific past experiences, leading to hopelessness&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Jiang|first=Wen|last2=Hu|first2=Guangtao|last3=Zhang|first3=Jingxuan|last4=Chen|first4=Ken|last5=Fan|first5=Dongni|last6=Feng|first6=Zhengzhi|date=2020-10-12|title=Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people|url=https://doi.org/10.1186/s12888-020-02877-6|journal=BMC Psychiatry|language=en|volume=20|issue=1|pages=501|doi=10.1186/s12888-020-02877-6|issn=1471-244X|pmc=7549224|pmid=33046032}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5}}&lt;/ref&gt;.

The current literature aims to shed light on a neglected niche of suicide research: autobiographical memory. Despite the overwhelming research suggesting correlations between OGM and depression and suicidal ideation, research has not thoroughly explored OGM's exact role in a cognitive, theoretical framework of suicidal ideation (specifically within the IMV model). By conducting a narrative review and integrating research on OGM's role in suicidal ideation, this paper furthers understanding on OGM's role in suicidal ideation within the IMV framework in high-risked population. 

==Mechanisms of OGM==
The inability to retrieve specific memories from one's autobiographical memory may lead to inefficient problem-solving abilities, which can impact one's ability to deal with difficult situations as they lack past experiences to rely on&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Arie|first=Miri|last2=Apter|first2=Alan|last3=Orbach|first3=Israel|last4=Yefet|first4=Yael|last5=Zalzman|first5=Gil|date=2008-01-01|title=Autobiographical memory, interpersonal problem solving, and suicidal behavior in adolescent inpatients|url=https://www.sciencedirect.com/science/article/pii/S0010440X07000922|journal=Comprehensive Psychiatry|volume=49|issue=1|pages=22–29|doi=10.1016/j.comppsych.2007.07.004|issn=0010-440X}}&lt;/ref&gt;. Failing to deal with difficult situations can drive an individual to hopelessness. In 1986, an article by Mark J. Williams and Keith Broadbent stipulated that individuals who recently attempted suicide had biased latencies in autobiographical memory retrieval and had reduced specificity in responses to especially positive cues&lt;ref&gt;{{Cite web|url=https://www.tandfonline.com/action/cookieAbsent|website=www.tandfonline.com|doi=10.1080/02699938808410925|access-date=2026-05-08}}&lt;/ref&gt;. OGM may not have a direct effect on suicide, but the evidence suggests that OGM intensifies the risk of suicidal ideation through deteriorating cognitive functioning.   

Rumination involves maladaptive dwelling on one's past negative emotions and feelings&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Treynor|first=Wendy|last2=Gonzalez|first2=Richard|last3=Nolen-Hoeksema|first3=Susan|date=2003-06-01|title=Rumination Reconsidered: A Psychometric Analysis|url=https://doi.org/10.1023/A:1023910315561|journal=Cognitive Therapy and Research|language=en|volume=27|issue=3|pages=247–259|doi=10.1023/A:1023910315561|issn=1573-2819}}&lt;/ref&gt;, and is associated with suicidal behavior/ideation&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational–volitional model of suicidal behaviour|url=https://royalsocietypublishing.org/doi/10.1098/rstb.2017.0268|journal=Philosophical Transactions of the Royal Society B: Biological Sciences|language=en|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=0962-8436|pmc=6053985|pmid=30012735}}&lt;/ref&gt;. One may reflect on their negative emotions and question such emotions in an abstract manner (&quot;How did I get to feel this way?&quot;&lt;ref name=&quot;:0&quot; /&gt;, &quot;Why did this happen to me?&quot;&lt;ref&gt;{{Cite journal|last=Sumner|first=Jennifer A.|date=2012-02|title=The mechanisms underlying overgeneral autobiographical memory: an evaluative review of evidence for the CaR-FA-X model|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3246105/|journal=Clinical Psychology Review|volume=32|issue=1|pages=34–48|doi=10.1016/j.cpr.2011.10.003|issn=1873-7811|pmc=3246105|pmid=22142837}}&lt;/ref&gt;), which causes one's memory retrieval to capture negative intermediate conceptual information (ex, &quot;I'm a failure&quot;) instead of specific memories. Repeated rumination strengthens these negative self-beliefs, leading to frequent capture of negative conceptual themes and impeding memory retrieval. This association of rumination and general memory aligns with the CaR-FA-X model&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;.  

Within the IMV model, we propose that OGM contributes to entrapment (a perceived sense of being trapped by defeat/humilitation) through impaired autobiographical memory, repeated rumination, and hopelessness. Positive factors, such as motivation to live, positive future thinking, and belongingess can offset the transition of entrapment → suicidal ideation, though negative factors, such as thwarted belongingness, very little social support, and perceived burdensomeness, may increase the chance of entrapment converting into suicidal ideation&lt;ref name=&quot;:1&quot; /&gt;.  

==OGM as a Vulnerability==
Evidence suggests that OGM is a cognitive vulnerability associated with depression and suicidal ideation, though its predictive relevance may vary depending on the population. 

A meta analysis performed by Sumner et. al (2010) found that OGM accounted for about 1-2% of the variance in depressive symptoms at follow-up&lt;ref name=&quot;:2&quot; /&gt;. A 2020 study found that OGM was associated with depressed patients' current suicidal ideation state and worse-point suicidal ideation, while OGM affected the healthy patients' worse-point suicidal ideation&lt;ref name=&quot;:2&quot; /&gt;. However, Crane et. al (2016) conducted a longitudinal study of n≈5800 adolescents from ages 13 to 16 and found that OGM was not significantly associated with depression and did not moderate the effect of life events, suggesting OGM may not be as generalizable to community samples vs. high-risk populations&lt;ref name=&quot;:2&quot; /&gt;. Even though the findings indicated that OGM was not significantly associated with depression, the study highlights that OGM may function as a vulnerability within high-risk populations. This suggests that OGM's predictive relevance are more prevalent in higher-risk populations. 

OGM persists even past depression, as found in Hallford et. al (2022). A meta-analysis indicated that participants with remitted depression continue to experience small to moderate deficiencies in being able to recall &quot;specific, event-level personal memories&quot;. This indicates that OGM isn't merely a symptom of depression, but may function as a risk factor for future depressive episodes&lt;ref name=&quot;:2&quot; /&gt;.    

Considering the above, the analysis suggests that OGM may function as a cognitive vulnerability that is further exacerbated in high-risk populations vs. the general population.    

=== OGM → Suicidal ideation ===
The research suggests that OGM is not just associated with depression, but is a contributing factor towards suicidal ideation. Jiang et. al (2020) found in a study of 365 participants, with roughly 51% of the participants clinically depressed while the other roughly 49% of participants were classified as &quot;healthy&quot;, that OGM had an increased presence in the depressed group vs. the &quot;healthy&quot; group. WSI (worst suicidal thoughts one has ever had [at a certain point]) and CSI (current point of suicidal ideation) were significantly affected by OGM in the depressed group. OGM was also found to be a mediator between CSI and childhood trauma in depressed patients. As OGM leads to negative memory biases and, therefore, the maintenance of a negative mental state, the researchers suggested that OGM may be a consistent contributor to suicidal ideation in depressed patients&lt;ref name=&quot;:3&quot; /&gt;. These findings are corroborated by a more recent (2025) study on depressed patients with varying levels of SI, where the researchers concluded that OGM may be &quot;a maladaptive cognitive avoidance strategy&quot; rather than simply a deterioration in memory. Zhu et. al (2025) further explain that individuals with OGM have a difficult time recalling positive memories, which reinforce negative recollections, spurring hopelessness and the transition to suicidal ideation&lt;ref&gt;{{Cite journal|last=Zhu|first=Ying|last2=Yin|first2=Qianlan|last3=Xu|first3=Huijing|last4=Xiao|first4=Fang|last5=Jiang|first5=Qian|last6=Liang|first6=Meng|last7=Cheng|first7=Qi|last8=Liu|first8=Taosheng|date=2025-11-24|title=Speech feature identification model for depressed individuals with suicidal ideation based on autobiographical memory|url=https://doi.org/10.1186/s12888-025-07635-0|journal=BMC Psychiatry|language=en|volume=25|issue=1|pages=1154|doi=10.1186/s12888-025-07635-0|issn=1471-244X|pmc=12713288|pmid=41286790}}&lt;/ref&gt;.

In a 2008 study on autobiographical memory, interpersonal problem-solving skills, and suicidal behaviour in adolescents and young adults, Arie et. al (2008) found that OGM was significantly associated with hopelessness and poor problem-solving abilities in adolescents. This suggests that being able to retrieve specific memories in one's autobiographical memory improves problem-solving skills, as they are able to draw back from past experiences to address challenging interpersonal situations&lt;ref name=&quot;:4&quot; /&gt;. Kaviani et. al (2011) found that depressed individuals with more severe suicidal ideation levels had more difficulty in retrieving specific thoughts in comparison to depressed individuals with less severe suicidal ideation&lt;ref&gt;{{Cite journal|last=Kaviani|first=H.|last2=Rahimi|first2=M.|last3=Rahimi-Darabad|first3=P.|last4=Naghavi|first4=K. Kamyar H.|date=2003|title=HOW AUTOBIOGRAPHICAL MEMORY DEFICITS AFFECT PROBLEM-SOLVING IN DEPRESSED PATIENTS|url=https://acta.tums.ac.ir/index.php/acta/article/view/2663|journal=Acta Medica Iranica|language=en-US|pages=194–198|issn=1735-9694}}&lt;/ref&gt;. These findings suggest that OGM plays a unique factor in contributing to suicidal ideation through maladaptive cognitive processing rather than being merely a symptom of depression.

=== Emphasis on High-Risk Population ===
Despite the importance of OGM and the findings indicating its potential amplification of suicidal ideation, OGM does not appear to be a consistent detriment in low-risk populations. 

Crane et. al (2016) conducted a longitudinal study of 5792 adolescents from ages 13 to 16 and found no significant findings that OGM played a direct or interactive role with depression, suicidal ideation, and self-harm (when accounted for confounding variables) in a general population&lt;ref&gt;{{Cite journal|last=Crane|first=Catherine|last2=Heron|first2=Jon|last3=Gunnell|first3=David|last4=Lewis|first4=Glyn|last5=Evans|first5=Jonathan|last6=Williams|first6=J. Mark G.|date=2016|title=Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort study|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/|journal=Memory (Hove, England)|volume=24|issue=3|pages=348–363|doi=10.1080/09658211.2015.1008014|issn=1464-0686|pmc=4743605|pmid=25716137}}&lt;/ref&gt;. The authors concluded that OGM appears to be more clinically meaningful in high-risk populations that are already cognitively vulnerable through depression and/or psychopathology. The OGM x Stress interaction theory is supported by another longitudinal study done on 174 Caucasian adolescents by Stange et. al (2012), where they found that OGM was found to be a vulnerability to adolescents with depression (especially emotional maltreatment)&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;. 

These findings, alongside with Jiang et. al (2020), suggest that OGM is context-dependent and may play a significant role in the development of depression and/or suicidal ideation if the individual is already susceptible for depression and/or mental disorders. This aligns with the diathesis-stress model, suggesting a set of factors interact with pre-existing vulnerabilities to produce a &quot;disordered state&quot;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5|pages=24-25}}&lt;/ref&gt;.

==Conclusion==
OGM has been found to be a cognitive vulnerability for suicidal ideation amongst clinically high-risk samples. Within the IMV model, OGM contributes to entrapment through impaired retrieval of past experiences, repeated rumination, and hopelessness. This can lead to negative Motivational Moderators (such as thwarted belongingness and perceived burdensomeness),and the transition from suicidal ideation to suicidal behavior within the Volitional Phase (where suicidal behavior is present). Despite significant research indicating OGM's unique contribution to suicidal ideation, OGM appears to be a significant contributor in high-risk/depressed populations. Future research should look into OGM's place in suicidal models and assess whether therapies targeting OGM, such as Memory Specificity Training (MEST), would be beneficial across diverse populations in reducing the risk of suicidal ideation. 

== References ==
{{Reflist}}
[[Category:Atcovi/OGM &amp; Suicide Poster]]</text>
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      <text bytes="17725" sha1="iqadvb8pr0ykl6mzw6weqp0q6lwehtz" xml:space="preserve">'''Major objective''': OGM may function as a context-dependent cognitive vulnerability that contributes to suicidal escalation through mechanisms embedded within the IMV model, particularly in high-risk populations. Through mechanisms outlined in the IMV framework, OGM may contribute to increased suicidal ideation within high-risk populations. 

==Introduction==
'''[[w:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory]]''' (OGM) describes a reduced ability to recall specific events in one's autobiographical memory. For example, one may remember attending a birthday party at some point in their life, but they could not uniquely recall a specific instance of attending a birthday party. OGM has been empirically associated with depression, with depressed individuals reporting higher levels of OGM than non-depressed individuals&lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Sumner|first=Jennifer A.|last2=Griffith|first2=James W.|last3=Mineka|first3=Susan|date=2010-07|title=Overgeneral autobiographical memory as a predictor of the course of depression: a meta-analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC2878838/|journal=Behaviour Research and Therapy|volume=48|issue=7|pages=614–625|doi=10.1016/j.brat.2010.03.013|issn=1873-622X|pmc=2878838|pmid=20399418}}&lt;/ref&gt;. Given the association of depression and suicidal ideation&lt;ref&gt;{{Cite journal|last=Chachamovich|first=Eduardo|last2=Stefanello|first2=Sabrina|last3=Botega|first3=Neury|last4=Turecki|first4=Gustavo|date=2009-05|title=[Which are the recent clinical findings regarding the association between depression and suicide?]|url=https://pubmed.ncbi.nlm.nih.gov/19565147|journal=Revista Brasileira De Psiquiatria (Sao Paulo, Brazil: 1999)|volume=31 Suppl 1|pages=S18–25|doi=10.1590/s1516-44462009000500004|issn=1516-4446|pmid=19565147}}&lt;/ref&gt;, utilizing the '''Integrated Motivational-Volitional (IMV) model''' provides a theoretical cognitive framework to argue that OGM intensifies suicidal escalation through moderators described in the model. 

The IMV model portrays suicidal behavior as an escalating, behavioral process divided into three phases: pre-motivational phase, motivational phase, and volitional phase. The motivational phase is characterized by suicidal ideation formation, where feelings of entrapment (described as a &quot;proximal [predictor] of suicidal ideation&quot;&lt;ref&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational-volitional model of suicidal behaviour|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/|journal=Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=1471-2970|pmc=6053985|pmid=30012735}}&lt;/ref&gt;), poor problem-solving abilities, brooding, and interpersonal vulnerabilities (thwarted belongingness and perceived burdensomeness) may transition the individual to the volitional phase. When looking at the IMV model and assessing where OGM contributes to suicidal ideation, OGM appears to impair problem-solving capabilities and the ability to learn from the past through reduced retrieval of specific past experiences, leading to hopelessness&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Jiang|first=Wen|last2=Hu|first2=Guangtao|last3=Zhang|first3=Jingxuan|last4=Chen|first4=Ken|last5=Fan|first5=Dongni|last6=Feng|first6=Zhengzhi|date=2020-10-12|title=Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people|url=https://doi.org/10.1186/s12888-020-02877-6|journal=BMC Psychiatry|language=en|volume=20|issue=1|pages=501|doi=10.1186/s12888-020-02877-6|issn=1471-244X|pmc=7549224|pmid=33046032}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5}}&lt;/ref&gt;.

The current literature aims to shed light on a neglected niche of suicide research: autobiographical memory. Despite the overwhelming research suggesting correlations between OGM and depression and suicidal ideation, research has not thoroughly explored OGM's exact role in a cognitive, theoretical framework of suicidal ideation (specifically within the IMV model). By conducting a narrative review and integrating research on OGM's role in suicidal ideation, this paper furthers understanding on OGM's role in suicidal ideation within the IMV framework in high-risk population. 

==Mechanisms of OGM==
The inability to retrieve specific memories from one's autobiographical memory may lead to inefficient problem-solving abilities, which can impact one's ability to deal with difficult situations as they lack past experiences to rely on&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Arie|first=Miri|last2=Apter|first2=Alan|last3=Orbach|first3=Israel|last4=Yefet|first4=Yael|last5=Zalzman|first5=Gil|date=2008-01-01|title=Autobiographical memory, interpersonal problem solving, and suicidal behavior in adolescent inpatients|url=https://www.sciencedirect.com/science/article/pii/S0010440X07000922|journal=Comprehensive Psychiatry|volume=49|issue=1|pages=22–29|doi=10.1016/j.comppsych.2007.07.004|issn=0010-440X}}&lt;/ref&gt;. Failing to deal with difficult situations can drive an individual to hopelessness. In 1986, an article by Mark J. Williams and Keith Broadbent stipulated that individuals who recently attempted suicide had biased latencies in autobiographical memory retrieval and had reduced specificity in responses to especially positive cues&lt;ref&gt;{{Cite web|url=https://www.tandfonline.com/action/cookieAbsent|website=www.tandfonline.com|doi=10.1080/02699938808410925|access-date=2026-05-08}}&lt;/ref&gt;. OGM may not have a direct effect on suicide, but the evidence suggests that OGM intensifies the risk of suicidal ideation through deteriorating cognitive functioning.   

Rumination involves maladaptive dwelling on one's past negative emotions and feelings&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Treynor|first=Wendy|last2=Gonzalez|first2=Richard|last3=Nolen-Hoeksema|first3=Susan|date=2003-06-01|title=Rumination Reconsidered: A Psychometric Analysis|url=https://doi.org/10.1023/A:1023910315561|journal=Cognitive Therapy and Research|language=en|volume=27|issue=3|pages=247–259|doi=10.1023/A:1023910315561|issn=1573-2819}}&lt;/ref&gt;, and is associated with suicidal behavior/ideation&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational–volitional model of suicidal behaviour|url=https://royalsocietypublishing.org/doi/10.1098/rstb.2017.0268|journal=Philosophical Transactions of the Royal Society B: Biological Sciences|language=en|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=0962-8436|pmc=6053985|pmid=30012735}}&lt;/ref&gt;. One may reflect on their negative emotions and question such emotions in an abstract manner (&quot;How did I get to feel this way?&quot;&lt;ref name=&quot;:0&quot; /&gt;, &quot;Why did this happen to me?&quot;&lt;ref&gt;{{Cite journal|last=Sumner|first=Jennifer A.|date=2012-02|title=The mechanisms underlying overgeneral autobiographical memory: an evaluative review of evidence for the CaR-FA-X model|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3246105/|journal=Clinical Psychology Review|volume=32|issue=1|pages=34–48|doi=10.1016/j.cpr.2011.10.003|issn=1873-7811|pmc=3246105|pmid=22142837}}&lt;/ref&gt;), which causes one's memory retrieval to capture negative intermediate conceptual information (ex, &quot;I'm a failure&quot;) instead of specific memories. Repeated rumination strengthens these negative self-beliefs, leading to frequent capture of negative conceptual themes and impeding memory retrieval. This association of rumination and general memory aligns with the CaR-FA-X model&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;.  

Within the IMV model, we propose that OGM contributes to entrapment (a perceived sense of being trapped by defeat/humilitation) through impaired autobiographical memory, repeated rumination, and hopelessness. Positive factors, such as motivation to live, positive future thinking, and belongingess can offset the transition of entrapment → suicidal ideation, though negative factors, such as thwarted belongingness, very little social support, and perceived burdensomeness, may increase the chance of entrapment converting into suicidal ideation&lt;ref name=&quot;:1&quot; /&gt;.  

==OGM as a Vulnerability==
Evidence suggests that OGM is a cognitive vulnerability associated with depression and suicidal ideation, though its predictive relevance may vary depending on the population. 

A meta analysis performed by Sumner et. al (2010) found that OGM accounted for about 1-2% of the variance in depressive symptoms at follow-up&lt;ref name=&quot;:2&quot; /&gt;. A 2020 study found that OGM was associated with depressed patients' current suicidal ideation state and worse-point suicidal ideation, while OGM affected the healthy patients' worse-point suicidal ideation&lt;ref name=&quot;:2&quot; /&gt;. However, Crane et. al (2016) conducted a longitudinal study of n≈5800 adolescents from ages 13 to 16 and found that OGM was not significantly associated with depression and did not moderate the effect of life events, suggesting OGM may not be as generalizable to community samples vs. high-risk populations&lt;ref name=&quot;:2&quot; /&gt;. Even though the findings indicated that OGM was not significantly associated with depression, the study highlights that OGM may function as a vulnerability within high-risk populations. This suggests that OGM's predictive relevance are prevalent in higher-risk populations. 

OGM persists even past depression, as found in Hallford et. al (2022). A meta-analysis indicated that participants with remitted depression continue to experience small to moderate deficiencies in being able to recall &quot;specific, event-level personal memories&quot;. This indicates that OGM isn't merely a symptom of depression, but may function as a risk factor for future depressive episodes&lt;ref name=&quot;:2&quot; /&gt;.    

Considering the above, the analysis suggests that OGM may function as a cognitive vulnerability that is further exacerbated in high-risk populations vs. the general population.    

=== OGM → Suicidal ideation ===
The research suggests that OGM is not just associated with depression, but is a contributing factor towards suicidal ideation. Jiang et. al (2020) found in a study of 365 participants, with roughly 51% of the participants clinically depressed while the other roughly 49% of participants were classified as &quot;healthy&quot;, that OGM had an increased presence in the depressed group vs. the &quot;healthy&quot; group. WSI (worst suicidal thoughts one has ever had [at a certain point]) and CSI (current point of suicidal ideation) were significantly affected by OGM in the depressed group. OGM was also found to be a mediator between CSI and childhood trauma in depressed patients. As OGM leads to negative memory biases and, therefore, the maintenance of a negative mental state, the researchers suggested that OGM may be a consistent contributor to suicidal ideation in depressed patients&lt;ref name=&quot;:3&quot; /&gt;. These findings are corroborated by a more recent (2025) study on depressed patients with varying levels of SI, where the researchers concluded that OGM may be &quot;a maladaptive cognitive avoidance strategy&quot; rather than simply a deterioration in memory. Zhu et. al (2025) further explain that individuals with OGM have a difficult time recalling positive memories, which reinforce negative recollections, spurring hopelessness and the transition to suicidal ideation&lt;ref&gt;{{Cite journal|last=Zhu|first=Ying|last2=Yin|first2=Qianlan|last3=Xu|first3=Huijing|last4=Xiao|first4=Fang|last5=Jiang|first5=Qian|last6=Liang|first6=Meng|last7=Cheng|first7=Qi|last8=Liu|first8=Taosheng|date=2025-11-24|title=Speech feature identification model for depressed individuals with suicidal ideation based on autobiographical memory|url=https://doi.org/10.1186/s12888-025-07635-0|journal=BMC Psychiatry|language=en|volume=25|issue=1|pages=1154|doi=10.1186/s12888-025-07635-0|issn=1471-244X|pmc=12713288|pmid=41286790}}&lt;/ref&gt;.

In a 2008 study on autobiographical memory, interpersonal problem-solving skills, and suicidal behaviour in adolescents and young adults, Arie et. al (2008) found that OGM was significantly associated with hopelessness and poor problem-solving abilities in adolescents. This suggests that being able to retrieve specific memories in one's autobiographical memory improves problem-solving skills, as they are able to draw back from past experiences to address challenging interpersonal situations&lt;ref name=&quot;:4&quot; /&gt;. Kaviani et. al (2011) found that depressed individuals with more severe suicidal ideation levels had more difficulty in retrieving specific thoughts in comparison to depressed individuals with less severe suicidal ideation&lt;ref&gt;{{Cite journal|last=Kaviani|first=H.|last2=Rahimi|first2=M.|last3=Rahimi-Darabad|first3=P.|last4=Naghavi|first4=K. Kamyar H.|date=2003|title=HOW AUTOBIOGRAPHICAL MEMORY DEFICITS AFFECT PROBLEM-SOLVING IN DEPRESSED PATIENTS|url=https://acta.tums.ac.ir/index.php/acta/article/view/2663|journal=Acta Medica Iranica|language=en-US|pages=194–198|issn=1735-9694}}&lt;/ref&gt;. These findings suggest that OGM plays a unique factor in contributing to suicidal ideation through maladaptive cognitive processing rather than being merely a symptom of depression.

=== Emphasis on High-Risk Population ===
Despite the importance of OGM and the findings indicating its potential amplification of suicidal ideation, OGM does not appear to be a consistent detriment in low-risk populations. 

Crane et. al (2016) conducted a longitudinal study of 5792 adolescents from ages 13 to 16 and found no significant findings that OGM played a direct or interactive role with depression, suicidal ideation, and self-harm (when accounted for confounding variables) in a general population&lt;ref&gt;{{Cite journal|last=Crane|first=Catherine|last2=Heron|first2=Jon|last3=Gunnell|first3=David|last4=Lewis|first4=Glyn|last5=Evans|first5=Jonathan|last6=Williams|first6=J. Mark G.|date=2016|title=Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort study|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/|journal=Memory (Hove, England)|volume=24|issue=3|pages=348–363|doi=10.1080/09658211.2015.1008014|issn=1464-0686|pmc=4743605|pmid=25716137}}&lt;/ref&gt;. The authors concluded that OGM appears to be more clinically meaningful in high-risk populations that are already cognitively vulnerable through depression and/or psychopathology. The OGM x Stress interaction theory is supported by another longitudinal study done on 174 Caucasian adolescents by Stange et. al (2012), where they found that OGM was found to be a vulnerability to adolescents with depression (especially emotional maltreatment)&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;. 

These findings, alongside with Jiang et. al (2020), suggest that OGM is context-dependent and may play a significant role in the development of depression and/or suicidal ideation if the individual is already susceptible for depression and/or mental disorders. This aligns with the diathesis-stress model, suggesting a set of factors interact with pre-existing vulnerabilities to produce a &quot;disordered state&quot;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5|pages=24-25}}&lt;/ref&gt;.

==Conclusion==
{{Notice|* reinforce your thesis
* clarify implications
* acknowledge limitations
* propose future directions}}
OGM has been found to be a cognitive vulnerability for suicidal ideation amongst clinically high-risk samples. Within the IMV model, OGM contributes to entrapment through impaired retrieval of past experiences, repeated rumination, and hopelessness. This can lead to negative Motivational Moderators (such as thwarted belongingness and perceived burdensomeness),and the transition from suicidal ideation to suicidal behavior within the Volitional Phase (where suicidal behavior is present). Despite significant research indicating OGM's unique contribution to suicidal ideation, OGM appears to be a significant contributor in high-risk/depressed populations. Future research should look into OGM's place in suicidal models and assess whether therapies targeting OGM, such as Memory Specificity Training (MEST), would be beneficial across diverse populations in reducing the risk of suicidal ideation. 

== References ==
{{Reflist}}
[[Category:Atcovi/OGM &amp; Suicide Poster]]</text>
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      <text bytes="17725" sha1="dam3aku37yjzkjdyqhzvpnr22gtcqh8" xml:space="preserve">'''Major objective''': OGM may function as a context-dependent cognitive vulnerability that contributes to suicidal escalation through mechanisms embedded within the IMV model, particularly in high-risk populations. Through mechanisms outlined in the IMV framework, OGM may contribute to increased suicidal ideation within high-risk populations. 

==Introduction==
'''[[w:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory]]''' (OGM) describes a reduced ability to recall specific events in one's autobiographical memory. For example, one may remember attending a birthday party at some point in their life, but they could not uniquely recall a specific instance of attending a birthday party. OGM has been empirically associated with depression, with depressed individuals reporting higher levels of OGM than non-depressed individuals&lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Sumner|first=Jennifer A.|last2=Griffith|first2=James W.|last3=Mineka|first3=Susan|date=2010-07|title=Overgeneral autobiographical memory as a predictor of the course of depression: a meta-analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC2878838/|journal=Behaviour Research and Therapy|volume=48|issue=7|pages=614–625|doi=10.1016/j.brat.2010.03.013|issn=1873-622X|pmc=2878838|pmid=20399418}}&lt;/ref&gt;. Given the association of depression and suicidal ideation&lt;ref&gt;{{Cite journal|last=Chachamovich|first=Eduardo|last2=Stefanello|first2=Sabrina|last3=Botega|first3=Neury|last4=Turecki|first4=Gustavo|date=2009-05|title=[Which are the recent clinical findings regarding the association between depression and suicide?]|url=https://pubmed.ncbi.nlm.nih.gov/19565147|journal=Revista Brasileira De Psiquiatria (Sao Paulo, Brazil: 1999)|volume=31 Suppl 1|pages=S18–25|doi=10.1590/s1516-44462009000500004|issn=1516-4446|pmid=19565147}}&lt;/ref&gt;, utilizing the '''Integrated Motivational-Volitional (IMV) model''' provides a theoretical cognitive framework to argue that OGM intensifies suicidal escalation through moderators described in the model. 

The IMV model portrays suicidal behavior as an escalating, behavioral process divided into three phases: pre-motivational phase, motivational phase, and volitional phase. The motivational phase is characterized by suicidal ideation formation, where feelings of entrapment (described as a &quot;proximal [predictor] of suicidal ideation&quot;&lt;ref&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational-volitional model of suicidal behaviour|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/|journal=Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=1471-2970|pmc=6053985|pmid=30012735}}&lt;/ref&gt;), poor problem-solving abilities, brooding, and interpersonal vulnerabilities (thwarted belongingness and perceived burdensomeness) may transition the individual to the volitional phase. When looking at the IMV model and assessing where OGM contributes to suicidal ideation, OGM appears to impair problem-solving capabilities and the ability to learn from the past through reduced retrieval of specific past experiences, leading to hopelessness&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Jiang|first=Wen|last2=Hu|first2=Guangtao|last3=Zhang|first3=Jingxuan|last4=Chen|first4=Ken|last5=Fan|first5=Dongni|last6=Feng|first6=Zhengzhi|date=2020-10-12|title=Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people|url=https://doi.org/10.1186/s12888-020-02877-6|journal=BMC Psychiatry|language=en|volume=20|issue=1|pages=501|doi=10.1186/s12888-020-02877-6|issn=1471-244X|pmc=7549224|pmid=33046032}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5}}&lt;/ref&gt;.

The current literature aims to shed light on a neglected niche of suicide research: autobiographical memory. Despite the overwhelming research suggesting correlations between OGM and depression and suicidal ideation, research has not thoroughly explored OGM's exact role in a cognitive, theoretical framework of suicidal ideation (specifically within the IMV model). By conducting a narrative review and integrating research on OGM's role in suicidal ideation, this paper furthers understanding on OGM's role in suicidal ideation within the IMV framework in high-risk population. 

==Mechanisms of OGM==
The inability to retrieve specific memories from one's autobiographical memory may lead to inefficient problem-solving abilities, which can impact one's ability to deal with difficult situations as they lack past experiences to rely on&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Arie|first=Miri|last2=Apter|first2=Alan|last3=Orbach|first3=Israel|last4=Yefet|first4=Yael|last5=Zalzman|first5=Gil|date=2008-01-01|title=Autobiographical memory, interpersonal problem solving, and suicidal behavior in adolescent inpatients|url=https://www.sciencedirect.com/science/article/pii/S0010440X07000922|journal=Comprehensive Psychiatry|volume=49|issue=1|pages=22–29|doi=10.1016/j.comppsych.2007.07.004|issn=0010-440X}}&lt;/ref&gt;. Failing to deal with difficult situations can drive an individual to hopelessness. In 1986, an article by Mark J. Williams and Keith Broadbent stipulated that individuals who recently attempted suicide had biased latencies in autobiographical memory retrieval and had reduced specificity in responses to especially positive cues&lt;ref&gt;{{Cite web|url=https://www.tandfonline.com/action/cookieAbsent|website=www.tandfonline.com|doi=10.1080/02699938808410925|access-date=2026-05-08}}&lt;/ref&gt;. OGM may not have a direct effect on suicide, but the evidence suggests that OGM intensifies the risk of suicidal ideation through deteriorating cognitive functioning.   

Rumination involves maladaptive dwelling on one's past negative emotions and feelings&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Treynor|first=Wendy|last2=Gonzalez|first2=Richard|last3=Nolen-Hoeksema|first3=Susan|date=2003-06-01|title=Rumination Reconsidered: A Psychometric Analysis|url=https://doi.org/10.1023/A:1023910315561|journal=Cognitive Therapy and Research|language=en|volume=27|issue=3|pages=247–259|doi=10.1023/A:1023910315561|issn=1573-2819}}&lt;/ref&gt;, and is associated with suicidal behavior/ideation&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational–volitional model of suicidal behaviour|url=https://royalsocietypublishing.org/doi/10.1098/rstb.2017.0268|journal=Philosophical Transactions of the Royal Society B: Biological Sciences|language=en|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=0962-8436|pmc=6053985|pmid=30012735}}&lt;/ref&gt;. One may reflect on their negative emotions and question such emotions in an abstract manner (&quot;How did I get to feel this way?&quot;&lt;ref name=&quot;:0&quot; /&gt;, &quot;Why did this happen to me?&quot;&lt;ref&gt;{{Cite journal|last=Sumner|first=Jennifer A.|date=2012-02|title=The mechanisms underlying overgeneral autobiographical memory: an evaluative review of evidence for the CaR-FA-X model|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3246105/|journal=Clinical Psychology Review|volume=32|issue=1|pages=34–48|doi=10.1016/j.cpr.2011.10.003|issn=1873-7811|pmc=3246105|pmid=22142837}}&lt;/ref&gt;), which causes one's memory retrieval to capture negative intermediate conceptual information (ex, &quot;I'm a failure&quot;) instead of specific memories. Repeated rumination strengthens these negative self-beliefs, leading to frequent capture of negative conceptual themes and impeding memory retrieval. This association of rumination and general memory aligns with the CaR-FA-X model&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;.  

Within the IMV model, we propose that OGM contributes to entrapment (a perceived sense of being trapped by defeat/humilitation) through impaired autobiographical memory, repeated rumination, and hopelessness. Positive factors, such as motivation to live, positive future thinking, and belongingess can offset the transition of entrapment → suicidal ideation, though negative factors, such as thwarted belongingness, very little social support, and perceived burdensomeness, may increase the chance of entrapment converting into suicidal ideation&lt;ref name=&quot;:1&quot; /&gt;.  

==OGM as a Vulnerability==
Evidence suggests that OGM is a cognitive vulnerability associated with depression and suicidal ideation, though its predictive relevance may vary depending on the population. 

A meta analysis performed by Sumner et. al (2010) found that OGM accounted for about 1-2% of the variance in depressive symptoms at follow-up&lt;ref name=&quot;:2&quot; /&gt;. A 2020 study found that OGM was associated with depressed patients' current suicidal ideation state and worse-point suicidal ideation, while OGM affected the healthy patients' worse-point suicidal ideation&lt;ref name=&quot;:2&quot; /&gt;. However, Crane et. al (2016) conducted a longitudinal study of n≈5800 adolescents from ages 13 to 16 and found that OGM was not significantly associated with depression and did not moderate the effect of life events, suggesting OGM may not be as generalizable to community samples vs. high-risk populations&lt;ref name=&quot;:2&quot; /&gt;. Even though the findings indicated that OGM was not significantly associated with depression, the study highlights that OGM may function as a vulnerability within high-risk populations. This suggests that OGM's predictive relevance are prevalent in higher-risk populations. 

OGM persists even past depression, as found in Hallford et. al (2022). A meta-analysis indicated that participants with remitted depression continue to experience small to moderate deficiencies in being able to recall &quot;specific, event-level personal memories&quot;. This indicates that OGM isn't merely a symptom of depression, but may function as a risk factor for future depressive episodes&lt;ref name=&quot;:2&quot; /&gt;.    

Considering the above, the analysis suggests that OGM may function as a cognitive vulnerability that is further exacerbated in high-risk populations vs. the general population.    

=== OGM → Suicidal ideation ===
The research suggests that OGM is not just associated with depression, but is a contributing factor towards suicidal ideation. Jiang et. al (2020) found in a study of 365 participants, with roughly 51% of the participants clinically depressed while the other roughly 49% of participants were classified as &quot;healthy&quot;, that OGM had an increased presence in the depressed group vs. the &quot;healthy&quot; group. WSI (worst suicidal thoughts one has ever had [at a certain point]) and CSI (current point of suicidal ideation) were significantly affected by OGM in the depressed group. OGM was also found to be a mediator between CSI and childhood trauma in depressed patients. As OGM leads to negative memory biases and, therefore, the maintenance of a negative mental state, the researchers suggested that OGM may be a consistent contributor to suicidal ideation in depressed patients&lt;ref name=&quot;:3&quot; /&gt;. These findings are corroborated by a more recent (2025) study on depressed patients with varying levels of SI, where the researchers concluded that OGM may be &quot;a maladaptive cognitive avoidance strategy&quot; rather than simply a deterioration in memory. Zhu et. al (2025) further explain that individuals with OGM have a difficult time recalling positive memories, which reinforce negative recollections, spurring hopelessness and the transition to suicidal ideation&lt;ref&gt;{{Cite journal|last=Zhu|first=Ying|last2=Yin|first2=Qianlan|last3=Xu|first3=Huijing|last4=Xiao|first4=Fang|last5=Jiang|first5=Qian|last6=Liang|first6=Meng|last7=Cheng|first7=Qi|last8=Liu|first8=Taosheng|date=2025-11-24|title=Speech feature identification model for depressed individuals with suicidal ideation based on autobiographical memory|url=https://doi.org/10.1186/s12888-025-07635-0|journal=BMC Psychiatry|language=en|volume=25|issue=1|pages=1154|doi=10.1186/s12888-025-07635-0|issn=1471-244X|pmc=12713288|pmid=41286790}}&lt;/ref&gt;.

In a 2008 study on autobiographical memory, interpersonal problem-solving skills, and suicidal behaviour in adolescents and young adults, Arie et. al (2008) found that OGM was significantly associated with hopelessness and poor problem-solving abilities in adolescents. This suggests that being able to retrieve specific memories in one's autobiographical memory improves problem-solving skills, as they are able to draw back from past experiences to address challenging interpersonal situations&lt;ref name=&quot;:4&quot; /&gt;. Kaviani et. al (2011) found that depressed individuals with more severe suicidal ideation levels had more difficulty in retrieving specific thoughts in comparison to depressed individuals with less severe suicidal ideation&lt;ref&gt;{{Cite journal|last=Kaviani|first=H.|last2=Rahimi|first2=M.|last3=Rahimi-Darabad|first3=P.|last4=Naghavi|first4=K. Kamyar H.|date=2003|title=HOW AUTOBIOGRAPHICAL MEMORY DEFICITS AFFECT PROBLEM-SOLVING IN DEPRESSED PATIENTS|url=https://acta.tums.ac.ir/index.php/acta/article/view/2663|journal=Acta Medica Iranica|language=en-US|pages=194–198|issn=1735-9694}}&lt;/ref&gt;. These findings suggest that OGM plays a unique factor in contributing to suicidal ideation through maladaptive cognitive processing rather than being merely a symptom of depression.

=== Emphasis on High-Risk Population ===
Despite the importance of OGM and the findings indicating its potential amplification of suicidal ideation, OGM does not appear to be a consistent detriment in low-risk populations. 

Crane et. al (2016) conducted a longitudinal study of 5792 adolescents from ages 13 to 16 and found no significant findings that OGM played a direct or interactive role with depression, suicidal ideation, and self-harm (when accounted for confounding variables) in a general population&lt;ref&gt;{{Cite journal|last=Crane|first=Catherine|last2=Heron|first2=Jon|last3=Gunnell|first3=David|last4=Lewis|first4=Glyn|last5=Evans|first5=Jonathan|last6=Williams|first6=J. Mark G.|date=2016|title=Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort study|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/|journal=Memory (Hove, England)|volume=24|issue=3|pages=348–363|doi=10.1080/09658211.2015.1008014|issn=1464-0686|pmc=4743605|pmid=25716137}}&lt;/ref&gt;. The authors concluded that OGM appears to be more clinically meaningful in high-risk populations that are already cognitively vulnerable through depression and/or psychopathology. The OGM x Stress interaction theory is supported by another longitudinal study done on 174 Caucasian adolescents by Stange et. al (2012), where they found that OGM was found to be a vulnerability to adolescents with depression (especially emotional maltreatment)&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;. 

These findings, alongside with Jiang et. al (2020), suggest that OGM is context-dependent and may play a significant role in the development of depression and/or suicidal ideation if the individual is already susceptible for depression and/or mental disorders. This aligns with the diathesis-stress model, suggesting a set of factors interact with pre-existing vulnerabilities to produce a &quot;disordered state&quot;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5|pages=24-25}}&lt;/ref&gt;.

==Conclusion==
{{Notice|
* reinforce your thesis
* clarify implications
* acknowledge limitations
* propose future directions}}
OGM has been found to be a cognitive vulnerability for suicidal ideation amongst clinically high-risk samples. Within the IMV model, OGM contributes to entrapment through impaired retrieval of past experiences, repeated rumination, and hopelessness. This can lead to negative Motivational Moderators (such as thwarted belongingness and perceived burdensomeness),and the transition from suicidal ideation to suicidal behavior within the Volitional Phase (where suicidal behavior is present). Despite significant research indicating OGM's unique contribution to suicidal ideation, OGM appears to be a significant contributor in high-risk/depressed populations. Future research should look into OGM's place in suicidal models and assess whether therapies targeting OGM, such as Memory Specificity Training (MEST), would be beneficial across diverse populations in reducing the risk of suicidal ideation.

== References ==
{{Reflist}}
[[Category:Atcovi/OGM &amp; Suicide Poster]]</text>
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      <text bytes="17674" sha1="2xonsvl88nvg3pipxx0ek1fgwu76poo" xml:space="preserve">'''Major objective''': OGM may function as a context-dependent cognitive vulnerability that contributes to suicidal escalation through mechanisms embedded within the IMV model, particularly in high-risk populations. Through mechanisms outlined in the IMV framework, OGM may contribute to increased suicidal ideation within high-risk populations. 

==Introduction==
'''[[w:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory]]''' (OGM) describes a reduced ability to recall specific events in one's autobiographical memory. For example, one may remember attending a birthday party at some point in their life, but they could not uniquely recall a specific instance of attending a birthday party. OGM has been empirically associated with depression, with depressed individuals reporting higher levels of OGM than non-depressed individuals&lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Sumner|first=Jennifer A.|last2=Griffith|first2=James W.|last3=Mineka|first3=Susan|date=2010-07|title=Overgeneral autobiographical memory as a predictor of the course of depression: a meta-analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC2878838/|journal=Behaviour Research and Therapy|volume=48|issue=7|pages=614–625|doi=10.1016/j.brat.2010.03.013|issn=1873-622X|pmc=2878838|pmid=20399418}}&lt;/ref&gt;. Given the association of depression and suicidal ideation&lt;ref&gt;{{Cite journal|last=Chachamovich|first=Eduardo|last2=Stefanello|first2=Sabrina|last3=Botega|first3=Neury|last4=Turecki|first4=Gustavo|date=2009-05|title=[Which are the recent clinical findings regarding the association between depression and suicide?]|url=https://pubmed.ncbi.nlm.nih.gov/19565147|journal=Revista Brasileira De Psiquiatria (Sao Paulo, Brazil: 1999)|volume=31 Suppl 1|pages=S18–25|doi=10.1590/s1516-44462009000500004|issn=1516-4446|pmid=19565147}}&lt;/ref&gt;, utilizing the '''Integrated Motivational-Volitional (IMV) model''' provides a theoretical cognitive framework to argue that OGM intensifies suicidal escalation through moderators described in the model. 

The IMV model portrays suicidal behavior as an escalating, behavioral process divided into three phases: pre-motivational phase, motivational phase, and volitional phase. The motivational phase is characterized by suicidal ideation formation, where feelings of entrapment (described as a &quot;proximal [predictor] of suicidal ideation&quot;&lt;ref&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational-volitional model of suicidal behaviour|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/|journal=Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=1471-2970|pmc=6053985|pmid=30012735}}&lt;/ref&gt;), poor problem-solving abilities, brooding, and interpersonal vulnerabilities (thwarted belongingness and perceived burdensomeness) may transition the individual to the volitional phase. When looking at the IMV model and assessing where OGM contributes to suicidal ideation, OGM appears to impair problem-solving capabilities and the ability to learn from the past through reduced retrieval of specific past experiences, leading to hopelessness&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Jiang|first=Wen|last2=Hu|first2=Guangtao|last3=Zhang|first3=Jingxuan|last4=Chen|first4=Ken|last5=Fan|first5=Dongni|last6=Feng|first6=Zhengzhi|date=2020-10-12|title=Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people|url=https://doi.org/10.1186/s12888-020-02877-6|journal=BMC Psychiatry|language=en|volume=20|issue=1|pages=501|doi=10.1186/s12888-020-02877-6|issn=1471-244X|pmc=7549224|pmid=33046032}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5}}&lt;/ref&gt;.

The current literature aims to shed light on a neglected niche of suicide research: autobiographical memory. Despite the overwhelming research suggesting correlations between OGM and depression and suicidal ideation, research has not thoroughly explored OGM's exact role in a cognitive, theoretical framework of suicidal ideation (specifically within the IMV model). By conducting a narrative review and integrating research on OGM's role in suicidal ideation, this paper furthers understanding on OGM's role in suicidal ideation within the IMV framework in high-risk population. 

==Mechanisms of OGM==
The inability to retrieve specific memories from one's autobiographical memory may lead to inefficient problem-solving abilities, which can impact one's ability to deal with difficult situations as they lack past experiences to rely on&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Arie|first=Miri|last2=Apter|first2=Alan|last3=Orbach|first3=Israel|last4=Yefet|first4=Yael|last5=Zalzman|first5=Gil|date=2008-01-01|title=Autobiographical memory, interpersonal problem solving, and suicidal behavior in adolescent inpatients|url=https://www.sciencedirect.com/science/article/pii/S0010440X07000922|journal=Comprehensive Psychiatry|volume=49|issue=1|pages=22–29|doi=10.1016/j.comppsych.2007.07.004|issn=0010-440X}}&lt;/ref&gt;. Failing to deal with difficult situations can drive an individual to hopelessness. In 1986, an article by Mark J. Williams and Keith Broadbent stipulated that individuals who recently attempted suicide had biased latencies in autobiographical memory retrieval and had reduced specificity in responses to especially positive cues&lt;ref&gt;{{Cite web|url=https://www.tandfonline.com/action/cookieAbsent|website=www.tandfonline.com|doi=10.1080/02699938808410925|access-date=2026-05-08}}&lt;/ref&gt;. OGM may not have a direct effect on suicide, but the evidence suggests that OGM intensifies the risk of suicidal ideation through deteriorating cognitive functioning.   

Rumination involves maladaptive dwelling on one's past negative emotions and feelings&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Treynor|first=Wendy|last2=Gonzalez|first2=Richard|last3=Nolen-Hoeksema|first3=Susan|date=2003-06-01|title=Rumination Reconsidered: A Psychometric Analysis|url=https://doi.org/10.1023/A:1023910315561|journal=Cognitive Therapy and Research|language=en|volume=27|issue=3|pages=247–259|doi=10.1023/A:1023910315561|issn=1573-2819}}&lt;/ref&gt;, and is associated with suicidal behavior/ideation&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational–volitional model of suicidal behaviour|url=https://royalsocietypublishing.org/doi/10.1098/rstb.2017.0268|journal=Philosophical Transactions of the Royal Society B: Biological Sciences|language=en|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=0962-8436|pmc=6053985|pmid=30012735}}&lt;/ref&gt;. One may reflect on their negative emotions and question such emotions in an abstract manner (&quot;How did I get to feel this way?&quot;&lt;ref name=&quot;:0&quot; /&gt;, &quot;Why did this happen to me?&quot;&lt;ref&gt;{{Cite journal|last=Sumner|first=Jennifer A.|date=2012-02|title=The mechanisms underlying overgeneral autobiographical memory: an evaluative review of evidence for the CaR-FA-X model|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3246105/|journal=Clinical Psychology Review|volume=32|issue=1|pages=34–48|doi=10.1016/j.cpr.2011.10.003|issn=1873-7811|pmc=3246105|pmid=22142837}}&lt;/ref&gt;), which causes one's memory retrieval to capture negative intermediate conceptual information (ex, &quot;I'm a failure&quot;) instead of specific memories. Repeated rumination strengthens these negative self-beliefs, leading to frequent capture of negative conceptual themes and impeding memory retrieval. This association of rumination and general memory aligns with the CaR-FA-X model&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;.  

Within the IMV model, I propose that OGM contributes to entrapment (a perceived sense of being trapped by defeat/humilitation) through impaired autobiographical memory, repeated rumination, and hopelessness. Positive factors, such as motivation to live, positive future thinking, and belongingess can offset the transition of entrapment → suicidal ideation, though negative factors from '''Motivational Moderators''' (MM), such as thwarted belongingness, very little social support, and perceived burdensomeness, may increase the chance of entrapment converting into suicidal ideation&lt;ref name=&quot;:1&quot; /&gt;.  

==OGM as a Vulnerability==
Evidence suggests that OGM is a cognitive vulnerability associated with depression and suicidal ideation, though its predictive relevance may vary depending on the population. 

A meta analysis performed by Sumner et. al (2010) found that OGM accounted for about 1-2% of the variance in depressive symptoms at follow-up&lt;ref name=&quot;:2&quot; /&gt;. A 2020 study found that OGM was associated with depressed patients' current suicidal ideation state and worse-point suicidal ideation, while OGM affected the healthy patients' worse-point suicidal ideation&lt;ref name=&quot;:2&quot; /&gt;. However, Crane et. al (2016) conducted a longitudinal study of n≈5800 adolescents from ages 13 to 16 and found that OGM was not significantly associated with depression and did not moderate the effect of life events, suggesting OGM may not be as generalizable to community samples vs. high-risk populations. OGM persists even past depression, as found in Hallford et. al (2022). A meta-analysis indicated that participants with remitted depression continue to experience small to moderate deficiencies in being able to recall &quot;specific, event-level personal memories&quot;. This indicates that OGM isn't merely a symptom of depression, but may function as a risk factor for future depressive episodes&lt;ref name=&quot;:2&quot; /&gt;. Even though the findings indicated that OGM was not significantly associated with depression, the study highlights that OGM may function as a vulnerability within high-risk populations. Across the scientific literature, it is evident that OGM has a notable factor in worsening suicidal ideation, but appears to be that OGM's predictive relevance is prevalent in higher-risk populations. 

=== OGM → Suicidal ideation ===
The research suggests that OGM is not just associated with depression, but is a contributing factor towards suicidal ideation. Jiang et. al (2020) found in a study of 365 participants, with roughly 51% of the participants clinically depressed while the other roughly 49% of participants were classified as &quot;healthy&quot;, that OGM had an increased presence in the depressed group vs. the &quot;healthy&quot; group. WSI (worst suicidal thoughts one has ever had [at a certain point]) and CSI (current point of suicidal ideation) were significantly affected by OGM in the depressed group. OGM was also found to be a mediator between CSI and childhood trauma in depressed patients. As OGM leads to negative memory biases and, therefore, the maintenance of a negative mental state, the researchers suggested that OGM may be a consistent contributor to suicidal ideation in depressed patients&lt;ref name=&quot;:3&quot; /&gt;. These findings are corroborated by a more recent (2025) study on depressed patients with varying levels of SI, where the researchers concluded that OGM may be &quot;a maladaptive cognitive avoidance strategy&quot; rather than simply a deterioration in memory. Zhu et. al (2025) further explain that individuals with OGM have a difficult time recalling positive memories, which reinforce negative recollections, spurring hopelessness and the transition to suicidal ideation&lt;ref&gt;{{Cite journal|last=Zhu|first=Ying|last2=Yin|first2=Qianlan|last3=Xu|first3=Huijing|last4=Xiao|first4=Fang|last5=Jiang|first5=Qian|last6=Liang|first6=Meng|last7=Cheng|first7=Qi|last8=Liu|first8=Taosheng|date=2025-11-24|title=Speech feature identification model for depressed individuals with suicidal ideation based on autobiographical memory|url=https://doi.org/10.1186/s12888-025-07635-0|journal=BMC Psychiatry|language=en|volume=25|issue=1|pages=1154|doi=10.1186/s12888-025-07635-0|issn=1471-244X|pmc=12713288|pmid=41286790}}&lt;/ref&gt;. 

In a 2008 study on autobiographical memory, interpersonal problem-solving skills, and suicidal behaviour in adolescents and young adults, Arie et. al (2008) found that OGM was significantly associated with hopelessness and poor problem-solving abilities in adolescents. This suggests that being able to retrieve specific memories in one's autobiographical memory improves problem-solving skills, as they are able to draw back from past experiences to address challenging interpersonal situations&lt;ref name=&quot;:4&quot; /&gt;. Kaviani et. al (2011) found that depressed individuals with more severe suicidal ideation levels had more difficulty in retrieving specific thoughts in comparison to depressed individuals with less severe suicidal ideation&lt;ref&gt;{{Cite journal|last=Kaviani|first=H.|last2=Rahimi|first2=M.|last3=Rahimi-Darabad|first3=P.|last4=Naghavi|first4=K. Kamyar H.|date=2003|title=HOW AUTOBIOGRAPHICAL MEMORY DEFICITS AFFECT PROBLEM-SOLVING IN DEPRESSED PATIENTS|url=https://acta.tums.ac.ir/index.php/acta/article/view/2663|journal=Acta Medica Iranica|language=en-US|pages=194–198|issn=1735-9694}}&lt;/ref&gt;. These findings suggest that OGM plays a unique factor in contributing to suicidal ideation through maladaptive cognitive processing rather than being merely a symptom of depression.

=== Emphasis on High-Risk Population ===
Despite the importance of OGM and the findings indicating its potential amplification of suicidal ideation, OGM does not appear to be a consistent detriment in low-risk populations. 

Crane et. al (2016) conducted a longitudinal study of 5792 adolescents from ages 13 to 16 and found no significant findings that OGM played a direct or interactive role with depression, suicidal ideation, and self-harm (when accounted for confounding variables) in a general population&lt;ref&gt;{{Cite journal|last=Crane|first=Catherine|last2=Heron|first2=Jon|last3=Gunnell|first3=David|last4=Lewis|first4=Glyn|last5=Evans|first5=Jonathan|last6=Williams|first6=J. Mark G.|date=2016|title=Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort study|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/|journal=Memory (Hove, England)|volume=24|issue=3|pages=348–363|doi=10.1080/09658211.2015.1008014|issn=1464-0686|pmc=4743605|pmid=25716137}}&lt;/ref&gt;. The authors concluded that OGM appears to be more clinically meaningful in high-risk populations that are already cognitively vulnerable through depression and/or psychopathology. The OGM x Stress interaction theory is supported by another longitudinal study done on 174 Caucasian adolescents by Stange et. al (2012), where they found that OGM was found to be a vulnerability to adolescents with depression (especially emotional maltreatment)&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;. 

These findings, alongside with Jiang et. al (2020), suggest that OGM is context-dependent and may play a significant role in the development of depression and/or suicidal ideation if the individual is already susceptible for depression and/or mental disorders. This aligns with the diathesis-stress model, suggesting a set of factors interact with pre-existing vulnerabilities to produce a &quot;disordered state&quot;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5|pages=24-25}}&lt;/ref&gt;.

==Conclusion==
{{Notice|
* reinforce your thesis
* clarify implications
* acknowledge limitations
* propose future directions}}
OGM has been found to be a cognitive vulnerability for suicidal ideation amongst clinically high-risk samples. Within the IMV model, OGM contributes to entrapment through impaired retrieval of past experiences, repeated rumination, and hopelessness. This can lead to negative Motivational Moderators (such as thwarted belongingness and perceived burdensomeness),and the transition from suicidal ideation to suicidal behavior within the Volitional Phase (where suicidal behavior is present). Despite significant research indicating OGM's unique contribution to suicidal ideation, OGM appears to be a significant contributor in high-risk/depressed populations. Future research should look into OGM's place in suicidal models and assess whether therapies targeting OGM, such as Memory Specificity Training (MEST), would be beneficial across diverse populations in reducing the risk of suicidal ideation.

== References ==
{{Reflist}}
[[Category:Atcovi/OGM &amp; Suicide Poster]]</text>
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      <text bytes="17710" sha1="ffou60li3ru073dsr2ojws0j5agyxy4" xml:space="preserve">'''Major objective''': OGM may function as a context-dependent cognitive vulnerability that contributes to suicidal escalation through mechanisms embedded within the IMV model, particularly in high-risk populations. Through mechanisms outlined in the IMV framework, OGM may contribute to increased suicidal ideation within high-risk populations. 

==Introduction==
'''[[w:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory]]''' (OGM) describes a reduced ability to recall specific events in one's autobiographical memory. For example, one may remember attending a birthday party at some point in their life, but they could not uniquely recall a specific instance of attending a birthday party. OGM has been empirically associated with depression, with depressed individuals reporting higher levels of OGM than non-depressed individuals&lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Sumner|first=Jennifer A.|last2=Griffith|first2=James W.|last3=Mineka|first3=Susan|date=2010-07|title=Overgeneral autobiographical memory as a predictor of the course of depression: a meta-analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC2878838/|journal=Behaviour Research and Therapy|volume=48|issue=7|pages=614–625|doi=10.1016/j.brat.2010.03.013|issn=1873-622X|pmc=2878838|pmid=20399418}}&lt;/ref&gt;. Given the association of depression and suicidal ideation&lt;ref&gt;{{Cite journal|last=Chachamovich|first=Eduardo|last2=Stefanello|first2=Sabrina|last3=Botega|first3=Neury|last4=Turecki|first4=Gustavo|date=2009-05|title=[Which are the recent clinical findings regarding the association between depression and suicide?]|url=https://pubmed.ncbi.nlm.nih.gov/19565147|journal=Revista Brasileira De Psiquiatria (Sao Paulo, Brazil: 1999)|volume=31 Suppl 1|pages=S18–25|doi=10.1590/s1516-44462009000500004|issn=1516-4446|pmid=19565147}}&lt;/ref&gt;, utilizing the '''Integrated Motivational-Volitional (IMV) model''' provides a theoretical cognitive framework to argue that OGM intensifies suicidal escalation through moderators described in the model. 

The IMV model portrays suicidal behavior as an escalating, behavioral process divided into three phases: pre-motivational phase, motivational phase, and volitional phase. The motivational phase is characterized by suicidal ideation formation, where feelings of entrapment (described as a &quot;proximal [predictor] of suicidal ideation&quot;&lt;ref&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational-volitional model of suicidal behaviour|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/|journal=Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=1471-2970|pmc=6053985|pmid=30012735}}&lt;/ref&gt;), poor problem-solving abilities, brooding, and interpersonal vulnerabilities (thwarted belongingness and perceived burdensomeness) may transition the individual to the volitional phase. When looking at the IMV model and assessing where OGM contributes to suicidal ideation, OGM appears to impair problem-solving capabilities and the ability to learn from the past through reduced retrieval of specific past experiences, leading to hopelessness&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Jiang|first=Wen|last2=Hu|first2=Guangtao|last3=Zhang|first3=Jingxuan|last4=Chen|first4=Ken|last5=Fan|first5=Dongni|last6=Feng|first6=Zhengzhi|date=2020-10-12|title=Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people|url=https://doi.org/10.1186/s12888-020-02877-6|journal=BMC Psychiatry|language=en|volume=20|issue=1|pages=501|doi=10.1186/s12888-020-02877-6|issn=1471-244X|pmc=7549224|pmid=33046032}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5}}&lt;/ref&gt;.

The current literature aims to shed light on a neglected niche of suicide research: autobiographical memory. Despite the overwhelming research suggesting correlations between OGM and depression and suicidal ideation, research has not thoroughly explored OGM's exact role in a cognitive, theoretical framework of suicidal ideation (specifically within the IMV model). By conducting a narrative review and integrating research on OGM's role in suicidal ideation, this paper furthers understanding on OGM's role in suicidal ideation within the IMV framework in high-risk population. 

==Mechanisms of OGM==
The inability to retrieve specific memories from one's autobiographical memory may lead to inefficient problem-solving abilities, which can impact one's ability to deal with difficult situations as they lack past experiences to rely on&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Arie|first=Miri|last2=Apter|first2=Alan|last3=Orbach|first3=Israel|last4=Yefet|first4=Yael|last5=Zalzman|first5=Gil|date=2008-01-01|title=Autobiographical memory, interpersonal problem solving, and suicidal behavior in adolescent inpatients|url=https://www.sciencedirect.com/science/article/pii/S0010440X07000922|journal=Comprehensive Psychiatry|volume=49|issue=1|pages=22–29|doi=10.1016/j.comppsych.2007.07.004|issn=0010-440X}}&lt;/ref&gt;. Failing to deal with difficult situations can drive an individual to hopelessness. In 1986, an article by Mark J. Williams and Keith Broadbent stipulated that individuals who recently attempted suicide had biased latencies in autobiographical memory retrieval and had reduced specificity in responses to especially positive cues&lt;ref&gt;{{Cite web|url=https://www.tandfonline.com/action/cookieAbsent|website=www.tandfonline.com|doi=10.1080/02699938808410925|access-date=2026-05-08}}&lt;/ref&gt;. OGM may not have a direct effect on suicide, but the evidence suggests that OGM intensifies the risk of suicidal ideation through deteriorating cognitive functioning.   

Rumination involves maladaptive dwelling on one's past negative emotions and feelings&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Treynor|first=Wendy|last2=Gonzalez|first2=Richard|last3=Nolen-Hoeksema|first3=Susan|date=2003-06-01|title=Rumination Reconsidered: A Psychometric Analysis|url=https://doi.org/10.1023/A:1023910315561|journal=Cognitive Therapy and Research|language=en|volume=27|issue=3|pages=247–259|doi=10.1023/A:1023910315561|issn=1573-2819}}&lt;/ref&gt;, and is associated with suicidal behavior/ideation&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational–volitional model of suicidal behaviour|url=https://royalsocietypublishing.org/doi/10.1098/rstb.2017.0268|journal=Philosophical Transactions of the Royal Society B: Biological Sciences|language=en|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=0962-8436|pmc=6053985|pmid=30012735}}&lt;/ref&gt;. One may reflect on their negative emotions and question such emotions in an abstract manner (&quot;How did I get to feel this way?&quot;&lt;ref name=&quot;:0&quot; /&gt;, &quot;Why did this happen to me?&quot;&lt;ref&gt;{{Cite journal|last=Sumner|first=Jennifer A.|date=2012-02|title=The mechanisms underlying overgeneral autobiographical memory: an evaluative review of evidence for the CaR-FA-X model|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3246105/|journal=Clinical Psychology Review|volume=32|issue=1|pages=34–48|doi=10.1016/j.cpr.2011.10.003|issn=1873-7811|pmc=3246105|pmid=22142837}}&lt;/ref&gt;), which causes one's memory retrieval to capture negative intermediate conceptual information (ex, &quot;I'm a failure&quot;) instead of specific memories. Repeated rumination strengthens these negative self-beliefs, leading to frequent capture of negative conceptual themes and impeding memory retrieval. This association of rumination and general memory aligns with the CaR-FA-X model&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;.  

Within the IMV model, I propose that OGM may contribute to entrapment (a perceived sense of being trapped by defeat/humilitation) through impaired autobiographical memory, repeated rumination, and hopelessness. Positive factors, such as motivation to live, positive future thinking, and belongingess can offset the transition of entrapment → suicidal ideation, though negative factors from '''Motivational Moderators''' (MM), such as thwarted belongingness, very little social support, and perceived burdensomeness, may increase the chance of entrapment converting into suicidal ideation&lt;ref name=&quot;:1&quot; /&gt;.  

==OGM as a Vulnerability==
Evidence suggests that OGM is a cognitive vulnerability associated with depression and suicidal ideation, though its predictive relevance may vary depending on the population. 

A meta analysis performed by Sumner et. al (2010) found that OGM accounted for about 1-2% of the variance in depressive symptoms at follow-up&lt;ref name=&quot;:2&quot; /&gt;. A 2020 study found that OGM was associated with depressed patients' current suicidal ideation state and worse-point suicidal ideation, while OGM affected the healthy patients' worse-point suicidal ideation&lt;ref name=&quot;:2&quot; /&gt;. However, Crane et. al (2016) conducted a longitudinal study of n≈5800 adolescents from ages 13 to 16 and found that OGM was not significantly associated with depression and did not moderate the effect of life events, suggesting OGM may not be as generalizable to community samples vs. high-risk populations. OGM persists even past depression, as found in Hallford et. al (2022). A meta-analysis indicated that participants with remitted depression continue to experience small to moderate deficiencies in being able to recall &quot;specific, event-level personal memories&quot;. This indicates that OGM isn't merely a symptom of depression, but may function as a risk factor for future depressive episodes&lt;ref name=&quot;:2&quot; /&gt;. Even though the findings indicated that OGM was not significantly associated with depression, the study highlights that OGM may function as a vulnerability within high-risk populations. Across the scientific literature, it is evident that OGM has a significant contributing factor in worsening suicidal ideation, but appears to be that OGM's predictive relevance is prevalent in higher-risk populations. 

=== OGM → Suicidal ideation ===
The research suggests that OGM may not just be associated with depression, but is may also be a contributing factor towards suicidal ideation. Jiang et. al (2020) found in a study of 365 participants, with roughly 51% of the participants clinically depressed while the other roughly 49% of participants were classified as &quot;healthy&quot;, that OGM had an increased presence in the depressed group vs. the &quot;healthy&quot; group. WSI (worst suicidal thoughts one has ever had [at a certain point]) and CSI (current point of suicidal ideation) were significantly affected by OGM in the depressed group. OGM was also found to be a mediator between CSI and childhood trauma in depressed patients. As OGM leads to negative memory biases and, therefore, the maintenance of a negative mental state, the researchers suggested that OGM may be a consistent contributor to suicidal ideation in depressed patients&lt;ref name=&quot;:3&quot; /&gt;. These findings are corroborated by a more recent (2025) study on depressed patients with varying levels of SI, where the researchers concluded that OGM may be &quot;a maladaptive cognitive avoidance strategy&quot; rather than simply a deterioration in memory. Zhu et. al (2025) further explain that individuals with OGM have a difficult time recalling positive memories, which reinforce negative recollections, spurring hopelessness and the transition to suicidal ideation&lt;ref&gt;{{Cite journal|last=Zhu|first=Ying|last2=Yin|first2=Qianlan|last3=Xu|first3=Huijing|last4=Xiao|first4=Fang|last5=Jiang|first5=Qian|last6=Liang|first6=Meng|last7=Cheng|first7=Qi|last8=Liu|first8=Taosheng|date=2025-11-24|title=Speech feature identification model for depressed individuals with suicidal ideation based on autobiographical memory|url=https://doi.org/10.1186/s12888-025-07635-0|journal=BMC Psychiatry|language=en|volume=25|issue=1|pages=1154|doi=10.1186/s12888-025-07635-0|issn=1471-244X|pmc=12713288|pmid=41286790}}&lt;/ref&gt;. 

In a 2008 study on autobiographical memory, interpersonal problem-solving skills, and suicidal behaviour in adolescents and young adults, Arie et. al (2008) found that OGM was significantly associated with hopelessness and poor problem-solving abilities in adolescents. This suggests that being able to retrieve specific memories in one's autobiographical memory improves problem-solving skills, as they are able to draw back from past experiences to address challenging interpersonal situations&lt;ref name=&quot;:4&quot; /&gt;. Kaviani et. al (2011) found that depressed individuals with more severe suicidal ideation levels had more difficulty in retrieving specific thoughts in comparison to depressed individuals with less severe suicidal ideation&lt;ref&gt;{{Cite journal|last=Kaviani|first=H.|last2=Rahimi|first2=M.|last3=Rahimi-Darabad|first3=P.|last4=Naghavi|first4=K. Kamyar H.|date=2003|title=HOW AUTOBIOGRAPHICAL MEMORY DEFICITS AFFECT PROBLEM-SOLVING IN DEPRESSED PATIENTS|url=https://acta.tums.ac.ir/index.php/acta/article/view/2663|journal=Acta Medica Iranica|language=en-US|pages=194–198|issn=1735-9694}}&lt;/ref&gt;. These findings suggest that OGM plays a unique factor in contributing to suicidal ideation through maladaptive cognitive processing rather than being merely a symptom of depression.

=== Emphasis on High-Risk Population ===
Despite the importance of OGM and the findings indicating its potential amplification of suicidal ideation, OGM does not appear to be a consistent detriment in low-risk populations. 

Crane et. al (2016) conducted a longitudinal study of 5792 adolescents from ages 13 to 16 and found no significant findings that OGM played a direct or interactive role with depression, suicidal ideation, and self-harm (when accounted for confounding variables) in a general population&lt;ref&gt;{{Cite journal|last=Crane|first=Catherine|last2=Heron|first2=Jon|last3=Gunnell|first3=David|last4=Lewis|first4=Glyn|last5=Evans|first5=Jonathan|last6=Williams|first6=J. Mark G.|date=2016|title=Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort study|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/|journal=Memory (Hove, England)|volume=24|issue=3|pages=348–363|doi=10.1080/09658211.2015.1008014|issn=1464-0686|pmc=4743605|pmid=25716137}}&lt;/ref&gt;. The authors concluded that OGM appears to be more clinically meaningful in high-risk populations that are already cognitively vulnerable through depression and/or psychopathology. The OGM x Stress interaction theory is supported by another longitudinal study done on 174 Caucasian adolescents by Stange et. al (2012), where they found that OGM was found to be a vulnerability to adolescents with depression (especially emotional maltreatment)&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;. 

These findings, alongside with Jiang et. al (2020), suggest that OGM is context-dependent and may play a significant role in the development of depression and/or suicidal ideation if the individual is already susceptible for depression and/or mental disorders. This aligns with the diathesis-stress model, suggesting a set of factors interact with pre-existing vulnerabilities to produce a &quot;disordered state&quot;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5|pages=24-25}}&lt;/ref&gt;.

==Conclusion==
{{Notice|
* reinforce your thesis
* clarify implications
* acknowledge limitations
* propose future directions}}
OGM has been found to be a cognitive vulnerability for suicidal ideation amongst clinically high-risk samples. Within the IMV model, OGM contributes to entrapment through impaired retrieval of past experiences, repeated rumination, and hopelessness. This can lead to negative Motivational Moderators (such as thwarted belongingness and perceived burdensomeness),and the transition from suicidal ideation to suicidal behavior within the Volitional Phase (where suicidal behavior is present). Despite significant research indicating OGM's unique contribution to suicidal ideation, OGM appears to be a significant contributor in high-risk/depressed populations. Future research should look into OGM's place in suicidal models and assess whether therapies targeting OGM, such as Memory Specificity Training (MEST), would be beneficial across diverse populations in reducing the risk of suicidal ideation.

== References ==
{{Reflist}}
[[Category:Atcovi/OGM &amp; Suicide Poster]]</text>
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      <text bytes="17719" sha1="13ecec9sp945wsv15fqeelv8dv5tkxm" xml:space="preserve">'''Major objective''': OGM may function as a context-dependent cognitive vulnerability that contributes to suicidal escalation through mechanisms embedded within the IMV model, particularly in high-risk populations. Through mechanisms outlined in the IMV framework, OGM may contribute to increased suicidal ideation within high-risk populations. 

==Introduction==
'''[[w:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory]]''' (OGM) describes a reduced ability to recall specific events in one's autobiographical memory. For example, one may remember attending a birthday party at some point in their life, but they could not uniquely recall a specific instance of attending a birthday party. OGM has been empirically associated with depression, with depressed individuals reporting higher levels of OGM than non-depressed individuals&lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Sumner|first=Jennifer A.|last2=Griffith|first2=James W.|last3=Mineka|first3=Susan|date=2010-07|title=Overgeneral autobiographical memory as a predictor of the course of depression: a meta-analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC2878838/|journal=Behaviour Research and Therapy|volume=48|issue=7|pages=614–625|doi=10.1016/j.brat.2010.03.013|issn=1873-622X|pmc=2878838|pmid=20399418}}&lt;/ref&gt;. Given the association of depression and suicidal ideation&lt;ref&gt;{{Cite journal|last=Chachamovich|first=Eduardo|last2=Stefanello|first2=Sabrina|last3=Botega|first3=Neury|last4=Turecki|first4=Gustavo|date=2009-05|title=[Which are the recent clinical findings regarding the association between depression and suicide?]|url=https://pubmed.ncbi.nlm.nih.gov/19565147|journal=Revista Brasileira De Psiquiatria (Sao Paulo, Brazil: 1999)|volume=31 Suppl 1|pages=S18–25|doi=10.1590/s1516-44462009000500004|issn=1516-4446|pmid=19565147}}&lt;/ref&gt;, utilizing the '''Integrated Motivational-Volitional (IMV) model''' provides a theoretical cognitive framework to argue that OGM intensifies suicidal escalation through moderators described in the model. 

The IMV model portrays suicidal behavior as an escalating, behavioral process divided into three phases: pre-motivational phase, motivational phase, and volitional phase. The motivational phase is characterized by suicidal ideation formation, where feelings of entrapment (described as a &quot;proximal [predictor] of suicidal ideation&quot;&lt;ref&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational-volitional model of suicidal behaviour|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/|journal=Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=1471-2970|pmc=6053985|pmid=30012735}}&lt;/ref&gt;), poor problem-solving abilities, brooding, and interpersonal vulnerabilities (thwarted belongingness and perceived burdensomeness) may transition the individual to the volitional phase. When looking at the IMV model and assessing where OGM contributes to suicidal ideation, OGM appears to impair problem-solving capabilities and the ability to learn from the past through reduced retrieval of specific past experiences, leading to hopelessness&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Jiang|first=Wen|last2=Hu|first2=Guangtao|last3=Zhang|first3=Jingxuan|last4=Chen|first4=Ken|last5=Fan|first5=Dongni|last6=Feng|first6=Zhengzhi|date=2020-10-12|title=Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people|url=https://doi.org/10.1186/s12888-020-02877-6|journal=BMC Psychiatry|language=en|volume=20|issue=1|pages=501|doi=10.1186/s12888-020-02877-6|issn=1471-244X|pmc=7549224|pmid=33046032}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5}}&lt;/ref&gt;.

The current literature aims to shed light on a neglected niche of suicide research: autobiographical memory. Despite the overwhelming research suggesting correlations between OGM and depression and suicidal ideation, research has not thoroughly explored OGM's exact role in a cognitive, theoretical framework of suicidal ideation (specifically within the IMV model). By conducting a narrative review and integrating research on OGM's role in suicidal ideation, this paper furthers understanding on OGM's role in suicidal ideation within the IMV framework in high-risk population. 

==Mechanisms of OGM==
The inability to retrieve specific memories from one's autobiographical memory may lead to inefficient problem-solving abilities, which can impact one's ability to deal with difficult situations as they lack past experiences to rely on&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Arie|first=Miri|last2=Apter|first2=Alan|last3=Orbach|first3=Israel|last4=Yefet|first4=Yael|last5=Zalzman|first5=Gil|date=2008-01-01|title=Autobiographical memory, interpersonal problem solving, and suicidal behavior in adolescent inpatients|url=https://www.sciencedirect.com/science/article/pii/S0010440X07000922|journal=Comprehensive Psychiatry|volume=49|issue=1|pages=22–29|doi=10.1016/j.comppsych.2007.07.004|issn=0010-440X}}&lt;/ref&gt;. Failing to deal with difficult situations can drive an individual to hopelessness. In 1986, an article by Mark J. Williams and Keith Broadbent stipulated that individuals who recently attempted suicide had biased latencies in autobiographical memory retrieval and had reduced specificity in responses to especially positive cues&lt;ref&gt;{{Cite web|url=https://www.tandfonline.com/action/cookieAbsent|website=www.tandfonline.com|doi=10.1080/02699938808410925|access-date=2026-05-08}}&lt;/ref&gt;. OGM may not have a direct effect on suicide, but the evidence suggests that OGM intensifies the risk of suicidal ideation through deteriorating cognitive functioning.   

Rumination involves maladaptive dwelling on one's past negative emotions and feelings&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Treynor|first=Wendy|last2=Gonzalez|first2=Richard|last3=Nolen-Hoeksema|first3=Susan|date=2003-06-01|title=Rumination Reconsidered: A Psychometric Analysis|url=https://doi.org/10.1023/A:1023910315561|journal=Cognitive Therapy and Research|language=en|volume=27|issue=3|pages=247–259|doi=10.1023/A:1023910315561|issn=1573-2819}}&lt;/ref&gt;, and is associated with suicidal behavior/ideation&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational–volitional model of suicidal behaviour|url=https://royalsocietypublishing.org/doi/10.1098/rstb.2017.0268|journal=Philosophical Transactions of the Royal Society B: Biological Sciences|language=en|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=0962-8436|pmc=6053985|pmid=30012735}}&lt;/ref&gt;. One may reflect on their negative emotions and question such emotions in an abstract manner (&quot;How did I get to feel this way?&quot;&lt;ref name=&quot;:0&quot; /&gt;, &quot;Why did this happen to me?&quot;&lt;ref&gt;{{Cite journal|last=Sumner|first=Jennifer A.|date=2012-02|title=The mechanisms underlying overgeneral autobiographical memory: an evaluative review of evidence for the CaR-FA-X model|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3246105/|journal=Clinical Psychology Review|volume=32|issue=1|pages=34–48|doi=10.1016/j.cpr.2011.10.003|issn=1873-7811|pmc=3246105|pmid=22142837}}&lt;/ref&gt;), which causes one's memory retrieval to capture negative intermediate conceptual information (ex, &quot;I'm a failure&quot;) instead of specific memories. Repeated rumination strengthens these negative self-beliefs, leading to frequent capture of negative conceptual themes and impeding memory retrieval. This association of rumination and general memory aligns with the CaR-FA-X model&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;.  

Within the IMV model, I propose that OGM may contribute to entrapment (a perceived sense of being trapped by defeat/humilitation) through impaired autobiographical memory, repeated rumination, and hopelessness. Positive factors, such as motivation to live, positive future thinking, and belongingess can offset the transition of entrapment → suicidal ideation, though negative factors from '''Motivational Moderators''' (MM), such as thwarted belongingness, very little social support, and perceived burdensomeness, may increase the chance of entrapment converting into suicidal ideation&lt;ref name=&quot;:1&quot; /&gt;.  

==OGM as a Vulnerability==
Evidence suggests that OGM is a cognitive vulnerability associated with depression and suicidal ideation, though its predictive relevance may vary depending on the population. 

A meta analysis performed by Sumner et. al (2010) found that OGM accounted for about 1-2% of the variance in depressive symptoms at follow-up&lt;ref name=&quot;:2&quot; /&gt;. A 2020 study found that OGM was associated with depressed patients' current suicidal ideation state and worse-point suicidal ideation, while OGM affected the healthy patients' worse-point suicidal ideation&lt;ref name=&quot;:2&quot; /&gt;. However, Crane et. al (2016) conducted a longitudinal study of n≈5800 adolescents from ages 13 to 16 and found that OGM was not significantly associated with depression and did not moderate the effect of life events, suggesting OGM may not be as generalizable to community samples vs. high-risk populations. OGM persists even past depression, as found in Hallford et. al (2022). A meta-analysis indicated that participants with remitted depression continue to experience small to moderate deficiencies in being able to recall &quot;specific, event-level personal memories&quot;. This indicates that OGM isn't merely a symptom of depression, but may function as a risk factor for future depressive episodes&lt;ref name=&quot;:2&quot; /&gt;. Even though the findings indicated that OGM was not significantly associated with depression, the study highlights that OGM may function as a vulnerability within high-risk populations. Across the scientific literature, OGM is suggested to have a significant contributing factor in worsening suicidal ideation, but appears to be have a more prevalent predictive relevance in higher-risk populations. 

=== OGM → Suicidal ideation ===
The research suggests that OGM may not just be associated with depression, but is may also be a contributing factor towards suicidal ideation. Jiang et. al (2020) found in a study of 365 participants, with roughly 51% of the participants clinically depressed while the other roughly 49% of participants were classified as &quot;healthy&quot;, that OGM had an increased presence in the depressed group vs. the &quot;healthy&quot; group. WSI (worst suicidal thoughts one has ever had [at a certain point]) and CSI (current point of suicidal ideation) were significantly affected by OGM in the depressed group. OGM was also found to be a mediator between CSI and childhood trauma in depressed patients. As OGM leads to negative memory biases and, therefore, the maintenance of a negative mental state, the researchers suggested that OGM may be a consistent contributor to suicidal ideation in depressed patients&lt;ref name=&quot;:3&quot; /&gt;. These findings are corroborated by a more recent (2025) study on depressed patients with varying levels of SI, where the researchers concluded that OGM may be &quot;a maladaptive cognitive avoidance strategy&quot; rather than simply a deterioration in memory. Zhu et. al (2025) further explain that individuals with OGM have a difficult time recalling positive memories, which reinforce negative recollections, spurring hopelessness and the transition to suicidal ideation&lt;ref&gt;{{Cite journal|last=Zhu|first=Ying|last2=Yin|first2=Qianlan|last3=Xu|first3=Huijing|last4=Xiao|first4=Fang|last5=Jiang|first5=Qian|last6=Liang|first6=Meng|last7=Cheng|first7=Qi|last8=Liu|first8=Taosheng|date=2025-11-24|title=Speech feature identification model for depressed individuals with suicidal ideation based on autobiographical memory|url=https://doi.org/10.1186/s12888-025-07635-0|journal=BMC Psychiatry|language=en|volume=25|issue=1|pages=1154|doi=10.1186/s12888-025-07635-0|issn=1471-244X|pmc=12713288|pmid=41286790}}&lt;/ref&gt;. 

In a 2008 study on autobiographical memory, interpersonal problem-solving skills, and suicidal behaviour in adolescents and young adults, Arie et. al (2008) found that OGM was significantly associated with hopelessness and poor problem-solving abilities in adolescents. This suggests that being able to retrieve specific memories in one's autobiographical memory improves problem-solving skills, as they are able to draw back from past experiences to address challenging interpersonal situations&lt;ref name=&quot;:4&quot; /&gt;. Kaviani et. al (2011) found that depressed individuals with more severe suicidal ideation levels had more difficulty in retrieving specific thoughts in comparison to depressed individuals with less severe suicidal ideation&lt;ref&gt;{{Cite journal|last=Kaviani|first=H.|last2=Rahimi|first2=M.|last3=Rahimi-Darabad|first3=P.|last4=Naghavi|first4=K. Kamyar H.|date=2003|title=HOW AUTOBIOGRAPHICAL MEMORY DEFICITS AFFECT PROBLEM-SOLVING IN DEPRESSED PATIENTS|url=https://acta.tums.ac.ir/index.php/acta/article/view/2663|journal=Acta Medica Iranica|language=en-US|pages=194–198|issn=1735-9694}}&lt;/ref&gt;. These findings suggest that OGM may play a unique factor in contributing to suicidal ideation through maladaptive cognitive processing rather than being merely a symptom of depression.

=== Emphasis on High-Risk Population ===
Despite the importance of OGM and the findings indicating its potential amplification of suicidal ideation, OGM does not appear to be a consistent detriment in low-risk populations. 

Crane et. al (2016) conducted a longitudinal study of 5792 adolescents from ages 13 to 16 and found no significant findings that OGM played a direct or interactive role with depression, suicidal ideation, and self-harm (when accounted for confounding variables) in a general population&lt;ref&gt;{{Cite journal|last=Crane|first=Catherine|last2=Heron|first2=Jon|last3=Gunnell|first3=David|last4=Lewis|first4=Glyn|last5=Evans|first5=Jonathan|last6=Williams|first6=J. Mark G.|date=2016|title=Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort study|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/|journal=Memory (Hove, England)|volume=24|issue=3|pages=348–363|doi=10.1080/09658211.2015.1008014|issn=1464-0686|pmc=4743605|pmid=25716137}}&lt;/ref&gt;. The authors concluded that OGM appears to be more clinically meaningful in high-risk populations that are already cognitively vulnerable through depression and/or psychopathology. The OGM x Stress interaction theory is supported by another longitudinal study done on 174 Caucasian adolescents by Stange et. al (2012), where they found that OGM was found to be a vulnerability to adolescents with depression (especially emotional maltreatment)&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;. 

These findings, alongside with Jiang et. al (2020), suggest that OGM is context-dependent and may play a significant role in the development of depression and/or suicidal ideation if the individual is already susceptible for depression and/or mental disorders. This aligns with the diathesis-stress model, suggesting a set of factors interact with pre-existing vulnerabilities to produce a &quot;disordered state&quot;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5|pages=24-25}}&lt;/ref&gt;.

==Conclusion==
{{Notice|
* reinforce your thesis
* clarify implications
* acknowledge limitations
* propose future directions}}
OGM has been found to be a cognitive vulnerability for suicidal ideation amongst clinically high-risk samples. Within the IMV model, OGM appears to contribute to entrapment through impaired retrieval of past experiences, repeated rumination, and hopelessness. This can lead to negative Motivational Moderators (such as thwarted belongingness and perceived burdensomeness),and the transition from suicidal ideation to suicidal behavior within the Volitional Phase (where suicidal behavior is present). Despite significant research indicating OGM's unique contribution to suicidal ideation, OGM appears to be a significant contributor in high-risk/depressed populations. Future research should look into OGM's place in suicidal models and assess whether therapies targeting OGM, such as Memory Specificity Training (MEST), would be beneficial across diverse populations in reducing the risk of suicidal ideation.

== References ==
{{Reflist}}
[[Category:Atcovi/OGM &amp; Suicide Poster]]</text>
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    <title>Athena problem</title>
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'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1. (You can draw this partial ordering as [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s)

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the set of such minimal elements must be a [[:w:Finite set|finite set]], e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), this set has exactly 77 [[:w:Element of a set|element]]s: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000.

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') (we call such families ''linear families''), and find the smallest prime &gt; ''b'' in all such families.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the set of the minimal elements in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem, we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'', in fact, Athena problem in base ''b'' covers the Sierpiński problem in base ''b'' and the Riesel problem in base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'', since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b''):

Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, which written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime).

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 prime (the largest of which has 2 digits): {11}

Base 3: 3 primes (the largest of which has 3 digits): {12, 21, 111}

Base 4: 5 primes (the largest of which has 3 digits): {11, 13, 23, 31, 221}

Base 5: 22 primes (the largest of which has 96 digits): {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013}

Base 6: 11 primes (the largest of which has 5 digits): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 primes (the largest of which has 17 digits): {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331}

Base 8: 75 primes (the largest of which has 221 digits): {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447}

Base 9: 151 primes (the largest of which has 1161 digits): {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011}

Base 10: 77 primes (the largest of which has 31 digits): {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}

Base 11: 1068 primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena problem base 11]

Base 12: 106 primes (the largest of which has 42 digits): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena problem base 13]

Base 14: 650 primes, the largest of which has 19699 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena problem base 14]

Base 15: 1284 primes, the largest of which has 157 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena problem base 15]

Base 16: 2347 primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena problem base 16]

Base 17: 10415 known primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of Athena problem base 17] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left17 Data of unsolved families for base 17]

Base 18: 549 primes, the largest of which has 6271 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena problem base 18]

Base 19: 31417 known primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of Athena problem base 19] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left19 Data of unsolved families for base 19]

Base 20: 3314 primes, the largest of which has 6271 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena problem base 20]

Base 21: 13386 known primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of Athena problem base 21] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left21 Data of unsolved families for base 21]

Base 22: 8003 primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena problem base 22]

Base 23: 65178 known primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of Athena problem base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for base 23]

Base 24: 3409 primes, the largest of which has 8134 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena problem base 24]

Base 25: 133639 known primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of Athena problem base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for base 25]

Base 26: 25256 known primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of Athena problem base 26]

Base 27: 102852 known primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of Athena problem base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for base 27]

Base 28: 25528 known primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 709070, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of Athena problem base 28]

Base 29: 355242 known primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of Athena problem base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for base 29]

Base 30: 2619 primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena problem base 30]

Base 31: 569323 known primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of Athena problem base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for base 31]

Base 32: 168882 known primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of Athena problem base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for base 32]

Base 33: 280012 known primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of Athena problem base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for base 33]

Base 34: 184785 known primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of Athena problem base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for base 34]

Base 35: 720002 known primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of Athena problem base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for base 35]

Base 36: 35286 known primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of Athena problem base 36]

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the minimal elements, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
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    <revision>
      <id>2810552</id>
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      <timestamp>2026-05-20T05:30:49Z</timestamp>
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      <text bytes="48268" sha1="1oq8cnbya0fmhcom6mqiwxlql1z7zcy" xml:space="preserve">{{mathematics}}
'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1. (You can draw this partial ordering as [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s)

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the set of such minimal elements must be a [[:w:Finite set|finite set]], e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), this set has exactly 77 [[:w:Element of a set|element]]s: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000.

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime &gt; ''b'' in all such families.

We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') &quot;linear&quot; families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our algorithm then proceeds as follows:

* 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''.
* 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by:
* 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''.
* 2.2. If possible, simplify the family.
* 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers &gt; ''b''), and if so then remove the family from ''L''.
* 3. Update ''L'', after each split examine the new families as in step 2.

e.g. in decimal (base ''b'' = 10):

''M'' := {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991}

''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L''

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L''

and since the family 3{0,3,6,9}3 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

etc.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the set of the minimal elements in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem, we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'', in fact, Athena problem in base ''b'' covers the Sierpiński problem in base ''b'' and the Riesel problem in base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'', since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b''):

Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, which written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime).

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 prime (the largest of which has 2 digits): {11}

Base 3: 3 primes (the largest of which has 3 digits): {12, 21, 111}

Base 4: 5 primes (the largest of which has 3 digits): {11, 13, 23, 31, 221}

Base 5: 22 primes (the largest of which has 96 digits): {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013}

Base 6: 11 primes (the largest of which has 5 digits): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 primes (the largest of which has 17 digits): {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331}

Base 8: 75 primes (the largest of which has 221 digits): {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447}

Base 9: 151 primes (the largest of which has 1161 digits): {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011}

Base 10: 77 primes (the largest of which has 31 digits): {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}

Base 11: 1068 primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena problem base 11]

Base 12: 106 primes (the largest of which has 42 digits): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena problem base 13]

Base 14: 650 primes, the largest of which has 19699 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena problem base 14]

Base 15: 1284 primes, the largest of which has 157 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena problem base 15]

Base 16: 2347 primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena problem base 16]

Base 17: 10415 known primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of Athena problem base 17] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left17 Data of unsolved families for base 17]

Base 18: 549 primes, the largest of which has 6271 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena problem base 18]

Base 19: 31417 known primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of Athena problem base 19] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left19 Data of unsolved families for base 19]

Base 20: 3314 primes, the largest of which has 6271 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena problem base 20]

Base 21: 13386 known primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of Athena problem base 21] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left21 Data of unsolved families for base 21]

Base 22: 8003 primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena problem base 22]

Base 23: 65178 known primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of Athena problem base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for base 23]

Base 24: 3409 primes, the largest of which has 8134 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena problem base 24]

Base 25: 133639 known primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of Athena problem base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for base 25]

Base 26: 25256 known primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of Athena problem base 26]

Base 27: 102852 known primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of Athena problem base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for base 27]

Base 28: 25528 known primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 709070, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of Athena problem base 28]

Base 29: 355242 known primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of Athena problem base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for base 29]

Base 30: 2619 primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena problem base 30]

Base 31: 569323 known primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of Athena problem base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for base 31]

Base 32: 168882 known primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of Athena problem base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for base 32]

Base 33: 280012 known primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of Athena problem base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for base 33]

Base 34: 184785 known primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of Athena problem base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for base 34]

Base 35: 720002 known primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of Athena problem base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for base 35]

Base 36: 35286 known primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of Athena problem base 36]

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the minimal elements, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
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      <id>2810553</id>
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      <text bytes="48285" sha1="hqx3wlsvkxw5kymw6stji378glokx6i" xml:space="preserve">{{mathematics}}
'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1. (You can draw this partial ordering as [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s)

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the set of such minimal elements must be a [[:w:Finite set|finite set]], e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), this set has exactly 77 [[:w:Element of a set|element]]s: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000.

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime &gt; ''b'' in all such families.

We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') &quot;linear&quot; families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows:

* 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''.
* 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by:
* 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''.
* 2.2. If possible, simplify the family.
* 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers &gt; ''b''), and if so then remove the family from ''L''.
* 3. Update ''L'', after each split examine the new families as in step 2.

e.g. in decimal (base ''b'' = 10):

''M'' := {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991}

''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L''

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L''

and since the family 3{0,3,6,9}3 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

etc.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the set of the minimal elements in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem, we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'', in fact, Athena problem in base ''b'' covers the Sierpiński problem in base ''b'' and the Riesel problem in base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'', since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b''):

Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, which written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime).

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 prime (the largest of which has 2 digits): {11}

Base 3: 3 primes (the largest of which has 3 digits): {12, 21, 111}

Base 4: 5 primes (the largest of which has 3 digits): {11, 13, 23, 31, 221}

Base 5: 22 primes (the largest of which has 96 digits): {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013}

Base 6: 11 primes (the largest of which has 5 digits): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 primes (the largest of which has 17 digits): {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331}

Base 8: 75 primes (the largest of which has 221 digits): {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447}

Base 9: 151 primes (the largest of which has 1161 digits): {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011}

Base 10: 77 primes (the largest of which has 31 digits): {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}

Base 11: 1068 primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena problem base 11]

Base 12: 106 primes (the largest of which has 42 digits): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena problem base 13]

Base 14: 650 primes, the largest of which has 19699 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena problem base 14]

Base 15: 1284 primes, the largest of which has 157 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena problem base 15]

Base 16: 2347 primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena problem base 16]

Base 17: 10415 known primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of Athena problem base 17] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left17 Data of unsolved families for base 17]

Base 18: 549 primes, the largest of which has 6271 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena problem base 18]

Base 19: 31417 known primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of Athena problem base 19] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left19 Data of unsolved families for base 19]

Base 20: 3314 primes, the largest of which has 6271 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena problem base 20]

Base 21: 13386 known primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of Athena problem base 21] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left21 Data of unsolved families for base 21]

Base 22: 8003 primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena problem base 22]

Base 23: 65178 known primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of Athena problem base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for base 23]

Base 24: 3409 primes, the largest of which has 8134 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena problem base 24]

Base 25: 133639 known primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of Athena problem base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for base 25]

Base 26: 25256 known primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of Athena problem base 26]

Base 27: 102852 known primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of Athena problem base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for base 27]

Base 28: 25528 known primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 709070, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of Athena problem base 28]

Base 29: 355242 known primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of Athena problem base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for base 29]

Base 30: 2619 primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena problem base 30]

Base 31: 569323 known primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of Athena problem base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for base 31]

Base 32: 168882 known primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of Athena problem base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for base 32]

Base 33: 280012 known primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of Athena problem base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for base 33]

Base 34: 184785 known primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of Athena problem base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for base 34]

Base 35: 720002 known primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of Athena problem base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for base 35]

Base 36: 35286 known primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of Athena problem base 36]

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the minimal elements, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
      <sha1>hqx3wlsvkxw5kymw6stji378glokx6i</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Candidates for Bureaucratship/Koavf</title>
    <ns>4</ns>
    <id>329564</id>
    <revision>
      <id>2810494</id>
      <parentid>2809360</parentid>
      <timestamp>2026-05-19T19:15:04Z</timestamp>
      <contributor>
        <username>Tommy Kronkvist</username>
        <id>31941</id>
      </contributor>
      <comment>/* Voting */ Support for user Koavf</comment>
      <origin>2810494</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2658" sha1="4eymqhqr02jusu1vb3kokon7ckkw69j" xml:space="preserve">=== {{User|Koavf}} ===
Per the [https://en.wikiversity.org/w/index.php?title=Wikiversity:Request_custodian_action&amp;oldid=2808455#Call_for_custodians_and_bureaucrats call] made by {{user|Jtneill}}, I am self-nominating for bureaucrat. I have had advanced user rights here for years and have been an bureaucrat on &lt;del&gt;[[:oversight:]]&lt;/del&gt;&lt;ins&gt;[[:outreach:]]&lt;/ins&gt; and a CheckUser on [[:species:]], among other wikis where I am an admin/sysop. I would be happy to help here as needed. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:48, 12 May 2026 (UTC)
==== Questions ====
:Can you clarify what you mean with the [[oversight:]] red link? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:21, 12 May 2026 (UTC)
:: Pinging [[User:Koavf|Koavf]] to my question. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:52, 14 May 2026 (UTC)
:::Whoops. It was [[:outreach:]]. I'm also a bureaucrat on [[:s:mul:]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:54, 14 May 2026 (UTC)
:::: No problem, thank you for the clarification. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:16, 14 May 2026 (UTC)

==== Comments ====
==== Voting ====
* {{support}} Prolific contributor to Wikimedia projects, with extensive and well-respected administrative experience and proven willingness to help English Wikiversity. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:41, 12 May 2026 (UTC)
* {{support}} [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:26, 13 May 2026 (UTC)
* {{support}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:16, 14 May 2026 (UTC)
* {{Support}}, as per Jtneill. [[User:Tommy Kronkvist|Tommy Kronkvist]] ([[User talk:Tommy Kronkvist|discuss]] • [[Special:Contributions/Tommy Kronkvist|contribs]]) 19:14, 19 May 2026 (UTC)

[[Category:Nominations for Bureaucratship|Koavf]]</text>
      <sha1>4eymqhqr02jusu1vb3kokon7ckkw69j</sha1>
    </revision>
    <revision>
      <id>2810528</id>
      <parentid>2810494</parentid>
      <timestamp>2026-05-20T02:43:02Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>/* Voting */ s</comment>
      <origin>2810528</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2806" sha1="hktz2hcc7igma1sdejkzsis53967q2o" xml:space="preserve">=== {{User|Koavf}} ===
Per the [https://en.wikiversity.org/w/index.php?title=Wikiversity:Request_custodian_action&amp;oldid=2808455#Call_for_custodians_and_bureaucrats call] made by {{user|Jtneill}}, I am self-nominating for bureaucrat. I have had advanced user rights here for years and have been an bureaucrat on &lt;del&gt;[[:oversight:]]&lt;/del&gt;&lt;ins&gt;[[:outreach:]]&lt;/ins&gt; and a CheckUser on [[:species:]], among other wikis where I am an admin/sysop. I would be happy to help here as needed. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:48, 12 May 2026 (UTC)
==== Questions ====
:Can you clarify what you mean with the [[oversight:]] red link? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:21, 12 May 2026 (UTC)
:: Pinging [[User:Koavf|Koavf]] to my question. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:52, 14 May 2026 (UTC)
:::Whoops. It was [[:outreach:]]. I'm also a bureaucrat on [[:s:mul:]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:54, 14 May 2026 (UTC)
:::: No problem, thank you for the clarification. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:16, 14 May 2026 (UTC)

==== Comments ====
==== Voting ====
* {{support}} Prolific contributor to Wikimedia projects, with extensive and well-respected administrative experience and proven willingness to help English Wikiversity. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:41, 12 May 2026 (UTC)
* {{support}} [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:26, 13 May 2026 (UTC)
* {{support}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:16, 14 May 2026 (UTC)
* {{Support}}, as per Jtneill. [[User:Tommy Kronkvist|Tommy Kronkvist]] ([[User talk:Tommy Kronkvist|discuss]] • [[Special:Contributions/Tommy Kronkvist|contribs]]) 19:14, 19 May 2026 (UTC)
* {{support}} no issues. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:43, 20 May 2026 (UTC)

[[Category:Nominations for Bureaucratship|Koavf]]</text>
      <sha1>hktz2hcc7igma1sdejkzsis53967q2o</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Candidates for Custodianship/PieWriter</title>
    <ns>4</ns>
    <id>329602</id>
    <revision>
      <id>2810525</id>
      <parentid>2810035</parentid>
      <timestamp>2026-05-20T02:39:19Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* Custodians offering mentorship */ Reply</comment>
      <origin>2810525</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2473" sha1="ezb9ymkotpwy9w2fpwnlhh2737ja7n6" xml:space="preserve">=== {{User|PieWriter}} ===
Hello everyone! I am submitting my request for temporary custodianship for 4 weeks following the recent notice seeking additional custodians. I believe I can contribute positively to the project.

If granted the tools, I would be able to block LTAs and spambots, who may try to disrupt the project. I already have curator rights, but they do not allow me to &lt;code&gt;block&lt;/code&gt; users. I would also be able to &lt;code&gt;undelete&lt;/code&gt; pages in order to help with the undeletion requests.

I already have experience working with advanced permissions on Wikiquote, where I am an administrator, so I am familiar with the responsibilities and expectations that come with administrator access. I understand the importance of using the tools carefully, and only when necessary.

Thanks for considering me :)

==== Custodians offering mentorship ====
* Hopefully someone else might step in here and mentor, but I am available. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:05, 14 May 2026 (UTC)
*:Sorry I skimmed passed this. I'm willing to mentor as I was willing to mentor beforehand in PieWriter's request for curatorship. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:39, 20 May 2026 (UTC)

==== Questions ====

==== Comments ====
* {{support}} PieWriter seems to know their way around wiki admin, has been contributing positively in this respect to Wikiversity, and is communicative. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:05, 14 May 2026 (UTC)
* {{comment}} See also: [[Wikiversity:Candidates for Curatorship/PieWriter]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:12, 14 May 2026 (UTC)
* {{support}} Seems safe enough for temporary adminship/custodianship and if it's successful and PW is motivated, I would encourage indefinite user rights. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:33, 14 May 2026 (UTC)
* {{support}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 02:53, 18 May 2026 (UTC)</text>
      <sha1>ezb9ymkotpwy9w2fpwnlhh2737ja7n6</sha1>
    </revision>
    <revision>
      <id>2810526</id>
      <parentid>2810525</parentid>
      <timestamp>2026-05-20T02:40:08Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* Comments */ s</comment>
      <origin>2810526</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2629" sha1="logaxbvg88p9nugudkjfoji9k3nrlmf" xml:space="preserve">=== {{User|PieWriter}} ===
Hello everyone! I am submitting my request for temporary custodianship for 4 weeks following the recent notice seeking additional custodians. I believe I can contribute positively to the project.

If granted the tools, I would be able to block LTAs and spambots, who may try to disrupt the project. I already have curator rights, but they do not allow me to &lt;code&gt;block&lt;/code&gt; users. I would also be able to &lt;code&gt;undelete&lt;/code&gt; pages in order to help with the undeletion requests.

I already have experience working with advanced permissions on Wikiquote, where I am an administrator, so I am familiar with the responsibilities and expectations that come with administrator access. I understand the importance of using the tools carefully, and only when necessary.

Thanks for considering me :)

==== Custodians offering mentorship ====
* Hopefully someone else might step in here and mentor, but I am available. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:05, 14 May 2026 (UTC)
*:Sorry I skimmed passed this. I'm willing to mentor as I was willing to mentor beforehand in PieWriter's request for curatorship. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:39, 20 May 2026 (UTC)

==== Questions ====

==== Comments ====
* {{support}} PieWriter seems to know their way around wiki admin, has been contributing positively in this respect to Wikiversity, and is communicative. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:05, 14 May 2026 (UTC)
* {{comment}} See also: [[Wikiversity:Candidates for Curatorship/PieWriter]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:12, 14 May 2026 (UTC)
* {{support}} Seems safe enough for temporary adminship/custodianship and if it's successful and PW is motivated, I would encourage indefinite user rights. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:33, 14 May 2026 (UTC)
* {{support}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 02:53, 18 May 2026 (UTC)
* {{support}} satisfactory work. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:40, 20 May 2026 (UTC)</text>
      <sha1>logaxbvg88p9nugudkjfoji9k3nrlmf</sha1>
    </revision>
  </page>
  <page>
    <title>Talk:WikiJournal Preprints/Using Wikidata to analyze the main historical trends in archaeological research on the Argentine continental Patagonian coast</title>
    <ns>1</ns>
    <id>329659</id>
    <revision>
      <id>2810431</id>
      <parentid>2809488</parentid>
      <timestamp>2026-05-19T13:52:44Z</timestamp>
      <contributor>
        <username>~2026-30168-33</username>
        <id>3078065</id>
      </contributor>
      <origin>2810431</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1558" sha1="jqs80lv2styyi01etrzc13cd7a3egnq" xml:space="preserve">
This work offers a comprehensive approach to the main historical trends in archaeological research on the Argentine Patagonian coast. In this sense, the study is very thorough, considering virtually all publications on the subject within a historiographical perspective focused on the development of archaeology in Argentina. A progressive development of the discipline is observed in diachronic terms. By uploading the information to Wikidata, creating an open database, the researchers identified trends based on different aspects, such as the topics addressed, geographical variations, and the representation of authors. Within this framework, for example, from the year 2000 onward, they recognize a process of consolidation and maturation of research in the archaeology of the continental Patagonian coast, which they correctly correlate with the implementation of expansionary policies that favored the development of Argentine science at both the national and international levels. On the other hand, and in contrast, it is also important to note that these findings highlight a significant decrease in the number of publications related to this topic in recent years. This correlates with the current situation of marked reductions in science funding in Argentina, with all the repercussions this has for research groups. As they rightly point out, this is a scenario in which a deliberate policy is resulting in a kind of &quot;scientific genocide&quot; that impacts all levels of the scientific system.
The manuscript can be published without modifications.</text>
      <sha1>jqs80lv2styyi01etrzc13cd7a3egnq</sha1>
    </revision>
  </page>
  <page>
    <title>User:Emmalinda1983</title>
    <ns>2</ns>
    <id>329737</id>
    <revision>
      <id>2810423</id>
      <timestamp>2026-05-19T13:07:14Z</timestamp>
      <contributor>
        <username>Emmalinda1983</username>
        <id>3077930</id>
      </contributor>
      <comment>Created basic details of Medical Billing Software</comment>
      <origin>2810423</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1614" sha1="rc1x30s0ebowdzaau0urqm7t9pchri7" xml:space="preserve">'''&lt;big&gt;Basics of Medical Billing Software&lt;/big&gt;'''

'''Introduction:'''

Medical billing software is used by healthcare providers to manage billing processes, insurance claims, patient payments, and revenue cycle operations. These systems help improve accuracy and reduce manual administrative work.

'''What is Medical Billing Software?'''

Medical billing software helps hospitals, clinics, and healthcare organizations automate billing and payment workflows. It is commonly used for claim submission, insurance verification, coding, and payment tracking.

&lt;big&gt;'''Key Features:'''&lt;/big&gt;

&lt;nowiki&gt;*&lt;/nowiki&gt; Insurance claim management

&lt;nowiki&gt;*&lt;/nowiki&gt; Payment tracking

&lt;nowiki&gt;*&lt;/nowiki&gt; Patient billing

&lt;nowiki&gt;*&lt;/nowiki&gt; Revenue cycle management

&lt;nowiki&gt;*&lt;/nowiki&gt; Reporting and analytics

&lt;nowiki&gt;*&lt;/nowiki&gt; Electronic health record integration

&lt;big&gt;'''Benefits:'''&lt;/big&gt; Medical billing software can help healthcare organizations:

&lt;nowiki&gt;*&lt;/nowiki&gt; Reduce billing errors

&lt;nowiki&gt;*&lt;/nowiki&gt; Improve operational efficiency

&lt;nowiki&gt;*&lt;/nowiki&gt; Speed up claim processing

&lt;nowiki&gt;*&lt;/nowiki&gt; Improve payment collection

&lt;nowiki&gt;*&lt;/nowiki&gt; Reduce paperwork

&lt;big&gt;'''Common Challenges:'''&lt;/big&gt; Some common challenges include:

&lt;nowiki&gt;*&lt;/nowiki&gt; Software integration issues

&lt;nowiki&gt;*&lt;/nowiki&gt; Data security concerns

&lt;nowiki&gt;*&lt;/nowiki&gt; Regulatory compliance

&lt;nowiki&gt;*&lt;/nowiki&gt; Staff training requirements

'''&lt;big&gt;Conclusion:&lt;/big&gt;'''

Medical billing software plays an important role in modern healthcare administration by improving billing accuracy, operational efficiency, and financial management.</text>
      <sha1>rc1x30s0ebowdzaau0urqm7t9pchri7</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:Emmalinda1983</title>
    <ns>3</ns>
    <id>329738</id>
    <revision>
      <id>2810426</id>
      <timestamp>2026-05-19T13:09:02Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* Welcome */ new section</comment>
      <origin>2810426</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2315" sha1="q8eq3y3yujd57viekdh9sakz2a37mqb" xml:space="preserve">==Welcome==
{{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]],  Emmalinda1983!'''|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Atcovi|me personally]] if you would like some [[Help:Contents|help]].

Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple.

We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies.

To find your way around, check out:
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Wikiversity:Introduction|Introduction to Wikiversity]]
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* [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;

* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]]
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&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]].

See you around Wikiversity! --—[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:09, 19 May 2026 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>q8eq3y3yujd57viekdh9sakz2a37mqb</sha1>
    </revision>
  </page>
  <page>
    <title>File:VLSI.Arith.2A.CLA.20260519.pdf</title>
    <ns>6</ns>
    <id>329739</id>
    <revision>
      <id>2810434</id>
      <timestamp>2026-05-19T14:05:42Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=Carry Lookahead Adders 2A traditional (20260519- 20260518)
|Source={{own|Young1lim}}
|Date=2026-05-19
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}</comment>
      <origin>2810434</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="281" sha1="6pfsw3jq82haz3l4owy8oepoduwksh3" xml:space="preserve">== Summary ==
{{Information
|Description=Carry Lookahead Adders 2A traditional (20260519- 20260518)
|Source={{own|Young1lim}}
|Date=2026-05-19
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== Licensing ==
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{{Information
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== Licensing ==
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{{Wikiversity:Bots/item|CommonsDelinker|Siebrand|prevention of broken image links|2022-05-23}}
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{{Wikiversity:Bots/item|MaintenanceBot|Dave Braunschweig|Various maintenance tasks.|2022-04-04}}
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{{Wikiversity:Bots/item|MediaWiki message delivery|MediaWiki message delivery|[[meta:MassMessage]] delivery|2022-05-31}}
{{Wikiversity:Bots/item|タチコマ robot|White Cat|Interwiki linking, double redirect fixing, commons delinking|2017-07-04&lt;ref&gt;Set to expire in 2022-07 per five years inactivity.&lt;/ref&gt;}}
{{Wikiversity:Bots/item|MGA73bot|MGA73|Working on files. See [[Wikiversity_talk:File_Review]] for background. Usually tasks are done with replace.py.|2023-05-04 to ????-??-??}}
{{Wikiversity:Bots/item|Leaderbot|Leaderboard|[[meta:Global reminder bot]]|N/A - it checks this wiki roughly every day, but only responds if needed}}
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Notes:
{{Reflist}}

The manually updated list above may not be current.  Check [http://en.wikiversity.org/w/index.php?title=Special%3AListusers&amp;group=bot&amp;username= bots group user list] which is complete and see contribs for each bot.

The [http://en.wikiversity.org/w/index.php?title=Special%3ALog&amp;type=makebot&amp;user=&amp;page=&amp;year=&amp;month=-1 bot status log] contains older entries of granting bot status.   Newer entries will appear in the [http://en.wikiversity.org/w/index.php?title=Special%3ALog&amp;type=rights&amp;user=&amp;page=&amp;year=&amp;month=-1 user rights log].</text>
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{{Wikiversity:Bots/item|MaintenanceBot|Dave Braunschweig|Various maintenance tasks.|2022-04-04}}
{{Wikiversity:Bots/item|MediaWiki default|MediaWiki default|setup edits in MediaWiki: namespace; the script is part of MediaWiki.&lt;ref&gt;[[User:MediaWiki default]] is the username used by a system maintenance script. It is ''not'' a bot and is unaffected by blocks.&lt;/ref&gt;|????-??-??&lt;ref&gt;Due to the bug described at [[phabricator:T36873]] the activity of this script no longer appears in Special:Contributions. It is not known when it was last run, or how often it has been used.&lt;/ref&gt;}}
{{Wikiversity:Bots/item|MediaWiki message delivery|MediaWiki message delivery|[[meta:MassMessage]] delivery|2022-05-31}}
{{Wikiversity:Bots/item|タチコマ robot|White Cat|Interwiki linking, double redirect fixing, commons delinking|2017-07-04&lt;ref&gt;Set to expire in 2022-07 per five years inactivity.&lt;/ref&gt;}}
{{Wikiversity:Bots/item|MGA73bot|MGA73|Working on files. See [[Wikiversity_talk:File_Review]] for background. Usually tasks are done with replace.py.|2023-05-04 to ????-??-??}}
{{Wikiversity:Bots/item|Leaderbot|Leaderboard|[[meta:Global reminder bot]]|N/A - it checks this wiki roughly every day, but only responds if needed}}
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Notes:
{{Reflist}}

The manually updated list above may not be current.  Check [http://en.wikiversity.org/w/index.php?title=Special%3AListusers&amp;group=bot&amp;username= bots group user list] which is complete and see contribs for each bot.

The [http://en.wikiversity.org/w/index.php?title=Special%3ALog&amp;type=makebot&amp;user=&amp;page=&amp;year=&amp;month=-1 bot status log] contains older entries of granting bot status.   Newer entries will appear in the [http://en.wikiversity.org/w/index.php?title=Special%3ALog&amp;type=rights&amp;user=&amp;page=&amp;year=&amp;month=-1 user rights log].</text>
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      <text bytes="817" sha1="awye9jd7qu44adthodwnd2o6zcfrac1" xml:space="preserve">This course assumes that the reader is somewhat familiar with the general principles of engineering, and has a strong basis in math. Otherwise, no other prior knowledge is required to follow this course.

The earth receives massive quantities of solar energy everyday, but despite its widespread abundance and regularity, the means to harvest the energy requires the conversion of light energy to other useful forms of energy. There are two primary energy needs that can be satisfied by solar energy; thermal needs and electrical needs.

Thermal solar can help satisfy the needs of heating and domestic hot water requirements, and additionally, includes the use of thermal energy for cooling purposes, such as cooling and solar air conditioning.

Solar photovoltaic (PV) can be used to help satisfy electrical demand.</text>
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      <text bytes="1641" sha1="iyo9quadxnbrba8mgxv0u44ad8rgkbq" xml:space="preserve">This course assumes that the reader is somewhat familiar with the general principles of engineering, and has a strong basis in math. Otherwise, no other prior knowledge is required to follow this course.

The earth receives massive quantities of solar energy everyday, but despite its widespread abundance and regularity, the means to harvest the energy requires the conversion of light energy to other useful forms of energy. There are two primary energy needs that can be satisfied by solar energy; thermal needs and electrical needs.

Thermal solar can help satisfy the needs of heating and domestic hot water requirements, and additionally, includes the use of thermal energy for cooling purposes, such as cooling and solar air conditioning.

Solar photovoltaic (PV) can be used to help satisfy electrical demand.

== Brief history in bullet points ==

* In 1839, the photovoltaic effect was discovered by the French physicist, Antoine-Cesar Becquerel (1788-1878). 
* In 1883, the first solar cell comprised of selenium, was created by the American Charles Fritts. The cell had an efficiency of less than 1%.
* In 1904 Albert Einstein (1879-1955) explained the photoelectric effect and for it, received the Nobel Prize in 1921.
* In 1954 Bell Laboratories (Charpin, Pearson and Price) introduced the first crystalline silicon cell that had an efficiency of 4%.
* In 1976, the first amorphous silicon cell was produced by RCA (D. Carlson and Wronski C.). These kinds of cells are considered 2nd generation.
* In 1991 Michael Graetzel and Brian O'Regan invented the Graetzel's cell at EPFL. These are considered 3rd generation solar cells.</text>
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== Licensing ==
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|Source={{own|Young1lim}}
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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|Source={{own|Young1lim}}
|Date=2026-05-20
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{{Information
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|Source={{own|Young1lim}}
|Date=2026-05-20
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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      <text bytes="2772" sha1="jo0oagxyq6hylqrn2zmkq53qlkzrfnz" xml:space="preserve">Ulrich Hoppe (ORCID https://orcid.org/0000-0002-8565-0349) is Professor of Audiology at the  [https://www.fau.eu/ Friedrich-Alexander University Erlangen-Nürnberg] at the ENT-clinic. He is head of the  Division of Audiology and the Cochlear Implant Centre CICERO. He is working as ad hoc reviewer for a number of audiological journals and is member of the editorial board of International Journal of Audiology, Medical Physics, Audiological Acoustics, and HNO. He is also member of the board of the European board of the [https://www.physicomedica-erlangen.de/index.shtml Societas Physico-Medica Erlangensis]  Association of Audiological Societies ([https://efas.ws/ EFAS]) and the German Audiological Society (Deutsche Gesellschaft für Audiologie, [https://dga-ev.com/ DGA]). He authored more than 160 publications listed at scopus and an H-index of 29.

=== Research Fields ===

* Hearing Aids and Hearing Aid Fitting
* Cochlear Implants
* Speech Audiometry
* Auditory Evoked Potentials
* Auditory Attention Decoding (AAD)
* Biosignal Processing
* Voice and Speech Signal Processing

=== Education ===
1987-1993 Studies in Medicine and Physics at the Georg-August-University, Göttingen

1993 Degree Diploma (M.Sc.) in Physics

1997 Doctorate (Ph.D.) in Electrical Engineering (Dr.-Ing., summa cum laude) at the Friedrich-Alexander-University Erlangen-Nürnberg; Technical faculty

Doctoral thesis: Hearing Loss estimation by means of auditory evoked potentials

2000 Doctorate (Ph.D.) in theoretical Medicine (Dr. rer. med.) University of Saarland; Medical faculty

Doctoral thesis: Wavelet-Analysis of transitory evoked otoacoustic emissions for differential diagnosis of cochlear hearing loss

2001 Habilitation/ Qualification as Associate Professor Friedrich-Alexander-University Erlangen-Nürnberg; Medical faculty; 

Post-doctoral thesis: Mechanisms of Hoarseness – Visualization and Interpretation by Means of Nonlinear Dynamics

=== Academic Positions and Training ===
1996 – 1999 Post Doctoral Fellowship, University of Saarland, Medical School; Department of Head and Neck Surgery; Audiological Laboratory

2000 - 2001  Post Doctoral Fellowship, University Hospital Erlangen; Department of Phoniatrics and Pediatric Audiology

2001 – 2003 Associate Professor, University Hospital Erlangen; Department of Phoniatrics and Pediatric Audiology

2003 – 2004 Full Professor of Biosignal Processing, Technical University of Ilmenau

Since 2004 Full Professor of Audiology and Head of the Division of Audiology University Hospital Erlangen; ENT Department

Since 2005 Member of the College of Engineering Sciences; University of Erlangen-Nuremberg

Since 2009 Head of the Cochlear Implant Centre Erlangen (CICERO) University Hospital Erlangen; ENT Department</text>
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      <comment>Welcome</comment>
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      <text bytes="2330" sha1="1j34nrpjxumgi87f9jp8hm2djayfto1" xml:space="preserve">==Welcome==
{{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]],  UlrichHoppe2023!'''|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Jtneill|me personally]] if you would like some [[Help:Contents|help]].

Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple.

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See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:38, 20 May 2026 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
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      <comment>Created page with &quot;== == Business Maintenance Plan == ==  === === Emergency Procedures === === • ''Contact local police''   • ''Contact hospital''   • ''Back up files every friday''   '''Important Reminder:'''  ''Always secure your customer data.''&quot;</comment>
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=== === Emergency Procedures === ===
• ''Contact local police'' 

• ''Contact hospital'' 

• ''Back up files every friday'' 

'''Important Reminder:'''  ''Always secure your customer data.''</text>
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